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Motion of hypersurfaces by Gauss curvature

September 2000
Pacific Journal of Mathematics 195(1)

September 2000
195(1)

DOI:10.2140/pjm.2000.195.1

Authors:

Australian National University

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n +2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-smooth initial hypersurfaces, and develop upper and lower bounds on the speed and the curvature independent of initial conditions. Applications are given to the flow by affine normal and to the existence of non-spherical homothetically contracting solutions.

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PACIFIC JOURNAL OF MATHEMATICS

Volume 195 No. 1 September 2000

Paciﬁc Journal of Mathematics 2000 Vol. 195, No. 1

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Volume 195 No. 1 September 2000

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PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

MOTION OF HYPERSURFACES BY GAUSS CURVATURE

Ben Andrews

We consider n-dimensional convex Euclidean hypersurfaces

moving with normal velocity proportional to a positive power

α of the Gauss curvature. We prove that hypersurfaces con-

tract to points in ﬁnite time, and for α ∈ (1/(n + 2], 1/n] we

also prove that in the limit the solutions evolve purely by ho-

mothetic contraction to the ﬁnal point. We prove existence

and uniqueness of solutions for non-smooth initial hypersur-

faces, and develop upper and lower bounds on the speed and

the curvature independent of i niti al conditions. Applications

are given to the ﬂow by aﬃne normal and to the exi stence of

non-spherical homothetically contracting solutions.

1. Introduction.

Motivation for the study of hypersurfaces moving by their Gauss curvature

comes from several sources:

1.1. Tumbling stones. W.J. Firey introduced the Gauss curvature ﬂow

in 1974, as a model of the wearing process undergone by a p ebble on a beach

[Fi]. Consider a stone which occupies an open, bounded convex region of

n+1

at time t = 0. The stone tumbles, and collides with a hyperplane (the

beach) with random orientation. We assume for simplicity that the amount

of material removed in a collision at a point x of the stone depends only on

the normal direction ν

(thus allowing some anisotropy in the material of

the stone). The number of collisions with a region B of the surface of the

stone is proportional to the measure of the set ν(B) = {ν

: x ∈ B} ⊂ S

of normal directions to B. This is equal to

(x), where K

is the

Gauss curvature of the hypersurface at x. The rate at which the stone wears

away at a point x is given by ρ(ν

for some positive function ρ on S

and we have the evolution equation

(1) ˙x = −ρ(ν

1.2. Aﬃne geometry: Inner parallel surfaces and the aﬃne normal

ﬂow. K. Leichtweiss introduced the notion of inner parallel surfaces for a

convex body in aﬃne geometry in the paper [Le]. Given a convex region Ω,

the idea is to construct a family of related regions P

Ω, by a procedure which

is well-deﬁned in the setting of aﬃne geometry — that is, if we perform

2 BEN ANDREWS

an area-preserving aﬃne transformation L to get a new region LΩ, then

(LΩ) = L(P

Ω).

The procedure is as follows: For each direction z ∈ S

, there exists a

unique supporting hyperplane H

= {hz, yi = h(z)} to Ω with normal di-

rection z pointing outward from Ω. The notion of parallel hyperplanes is

well-deﬁned in aﬃne geometry, as is the notion of the volume of a region.

Hence we can choose a unique hyperplane H

z,t

parallel to H

such that the

volume of the part of Ω between H

z,t

and H

is equal to t

(n+2)/2

(in the case

where Ω is the region above a paraboloid, this choice of exponent ensures

that H

z,t

moves at constant speed). Equivalently, we can deﬁne a function

(z) by the requirement

(2) Vol ({y ∈ Ω : h

(z) ≤ hy, zi ≤ h(z)}) = t

(n+2)/2

Then we deﬁne P

Ω to be the convex set

(3)

z∈S

{y ∈ R

n+1

: hz, yi ≤ h

(z)}.

In contrast to the corresponding situation in Euclidean geometry, this

procedure does not deﬁne a semi-group: If we begin with a region Ω, con-

struct the regions P

Ω, and use them to construct the regions P

Ω, then

these are not in general given by P

Ω for any t

. To remedy this we consider

t/n

Ω, obtained by following the above construction repeatedly over small

intervals, and take the limit n → ∞ of inﬁnitesimally small steps to obtain

a region

Ω. This deﬁnes a deformation which is clearly well-deﬁned in

aﬃne geometry, and satisﬁes the semi-group property

t+τ

To ﬁnd an explicit description of this deformation in the case where Ω

is smooth and strictly convex, we consider the regions P

Ω in the limit of

small t: Fix z, and choose coordinates for R

n+1

such that e

, . . . , e

span the

supporting hyperplane H

of Ω, and the supporting point is at the origin.

Then M = ∂Ω is locally a graph in these coordinates:

n+1

= −

i,j=1

+ O(|x|

where h

is the second fundamental form at the supporting point. There ex-

ists a volume-preserving linear transformation which ﬁxes the e

n+1

direction

and brings M locally to the form

n+1

= −

1/n

i=1

+ O(|x|

)

where K = det h

is the Gauss curvature at the supporting point. Then

Vol ({y ∈ Ω : h(z) − d ≤ hy, zi ≤ h(z)})

= 2

n/2

−1/2

(n+2)/2

+ O(d

(n+3)/2

)

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 3

where ω

is the volume of the unit ball in R

, and hence the requirement

(2) implies that

(z) = h(z) −

n+2

t + O(t

3/2

It follows that the limiting deformation is given by the equation

(4) ˙x = −c

n+2

This evolution equation is the simplest invariant ﬂow in aﬃne diﬀerential

geometry; up to reparametrisation it is the motion of a hypersurface in the

direction of its aﬃ ne normal vector. This has been considered in [ST1, ST2]

for the case of convex curves in the plane, and in [A4] more generally. For

nonconvex curves results were recently obtained in [AST].

1.3. Image analysis. Many fundamental problems in image analysis have

been approached using geometric ﬂows: An image represented by a grey-

scale density function u can be processed to remove noise by smoothing the

level sets of u with a parabolic ﬂow. Various candidates have been con-

sidered, but in [AGLM] axioms were proposed which included the natural

requirement of aﬃne invariance. This leads to the evolution Equation (4).

In the case of nonconvex hypersurfaces this is no longer parabolic, and var-

ious authors (see [AGLM], [CS], [NK]) have considered the generalization

(for two-dimensional surfaces)

˙x = −(sgnH) max{K, 0}

1/4

where H is the mean curvature. Applications of plane curve evolution equa-

tions to image analysis and computer vision are described in [AGLM],

[OST], and [ST1]-[ST3].

1.4. Gradient ﬂows of the mean width. The width of a convex region

Ω in a direction z ∈ S

is deﬁned by

w(z) = sup

∈Ω

− y

, zi = h(z) + h(−z)

where h(z) = sup

y∈Ω

hy, zi is the support function of Ω. The mean width

(Ω, ϕ) with resp e ct to a measure ϕdµ on S

is given (up to a constant

factor) by integrating the width over all directions z ∈ S

(Ω) =

w(z)ϕ(z)dµ(z) =

h(z)(ϕ(z) + ϕ(−z))dµ(z)

∂Ω

Kh ˜ϕ dH

4 BEN ANDREWS

where ˜ϕ(z) = ϕ(z) + ϕ(−z). The ﬁrst variation formula for the mean width

can be calculated as follows: Consider a smooth family Ω

of convex regions,

with support functions h

(z) such that

∂

∂t

(z)



t=0

= f(z),

then

(Ω

)



t=0

˜ϕ(z)f(z) dµ =

∂Ω

Kf ˜ϕ dH

We consider the ﬂow of steepest descent of the mean width in L

spaces

on ∂Ω — that is, we seek that variation f for which V

(Ω

, ϕ) decreases

fastest amongst all variations with the same L

norm



∂Ω

|f|

σ(ν) dH



1/p

(σ is a positive smooth function on S

): By the H¨older inequality we have

for p > 1



∂Ω

Kf ˜ϕdH



∂Ω

˜ϕ

σdH



≤



∂Ω

|f|

σdH



1/p

∂Ω



K ˜ϕ



p−1

σdH

1−1/p

with equality if and only if f = c (K ˜ϕ/σ)

1/(p−1)

. The ﬂow of steepes t descent

is therefore

(5) ˙x = −ρ(ν

1/(p−1)

where ρ = ( ˜ϕ/σ)

1/(p−1)

is a smooth positive function on S

1.5. Evolving hypersurfaces and degenerate fully nonlinear PDE.

The evolution equations derived above are included in a large class of par-

abolic evolution equations for hypersurfaces which have been considered

before. Simplest in this class is the mean curvature ﬂow, in which a hy-

persurface moves in the direction of its inward normal with speed given

by the mean curvature. Huisken [Hu] showed that convex hypersurfaces

moving under such equations contract to points in ﬁnite time, and that the

hypersurfaces become s pherical in shape in the process . This argument has

since been extended to many processes where convex hypersurfaces move

with speeds given by homogeneous degree one, concave or convex monotone

symmetric functions of the principal curvatures: Chow considered ﬂows by

the nth root of the Gauss curvature [Ch1] and the square root of the scalar

curvature [Ch2], and the author has considered a general class of s uch evo-

lution equations [A1]. Corresponding results for ﬂows where the speed has

other positive degrees of homogeneity in the curvature seem much harder to

prove. The author has treated the special case of ﬂow by the power 1/(n+2)

of the Gauss curvature, which is the ﬂow by aﬃne normal [A4]. Tso [Ts]

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 5

and Chow [Ch1] have shown that hypersurfaces moving with speed equal to

any positive power of the Gauss curvature contract to points in ﬁnite time.

The Gauss curvature ﬂows form a convenient class of examples of par-

abolic equations with varying degeneracy: For large α they become more

degenerate, and for small α they become singularly parabolic. Intermediate

values of α are singular in some situations and degenerate in others. The

precise eﬀect of such degeneracy or singularity on the regularity of solutions

is extremely complicated. In particular, it would be interesting to know

how irregular solutions can be, how the regularity estimates depend on time

(particularly where the initial solution is highly irregular), and how solutions

behave in the neighbourhood of degenerate or singular regions.

There are several other important families of PDE for which similar ques-

tions can be asked — in particular, natural families of parabolic equations

with varying degeneracy include the porous medium equations

˙u = ∆(|u|

m−1

u),

and the p-harmonic heat ﬂows

˙u = ∇ ·



|∇u|

p−2

∇u



for which there is also a natural generalization to p-harmonic maps between

Riemannian manifolds. The Gauss curvature ﬂows can be considered a

geometric analogue of the porous medium equations.

In the case of curves in the plane, more complete results are known:

Gage [Ga1]-[Ga2] and Hamilton [GH] showed that convex curves contract

to points in ﬁnite time and become round under the curve shortening ﬂow

(where the speed of motion equals the curvature), and Grayson [Gr] ex-

tended this by showing that any embedded curve eventually becomes convex.

This was extended to include anisotropic analogues of the curve-shortening

ﬂow by Gage [Ga3] in the convex case, and by Oaks [Oa] for nonconvex

curves. The author considered equations of varying degeneracy in the con-

vex case [A2], [A8], and obtained optimal estimates on the regularity of

solutions, including their initial behaviour. The particular case of the aﬃne

normal ﬂow has also been extended to nonconvex curves [AST].

2. The result.

Our main aim in this paper is to prove results about the regularity and

limiting behaviour of solutions of the Gauss curvature ﬂows of the form

(6)

= −ρ(ν(x))K(x)

ν(x),

where α is in the range (1/(n + 2), 1/n]. These particular exponents arise

as follows in the proof: We ﬁrst prove (in Section 4 of the paper) that the

solutions of Gauss curvature ﬂows have isoperimetric ratio bounded as long

6 BEN ANDREWS

as they exist, provided α is greater than the critical value 1/(n + 2). This

value is sharp — the ﬂow with ρ ≡ 1 and α = 1/(n + 2) is the aﬃne normal

ﬂow, for which solutions converge to ellipsoids of arbitrary eccentricity [A4],

and so the isoperimetric ratio tends to stay bounded but does not generally

improve; in a separate paper [A9] we prove that for exponents smaller than

this (or equal to this if ρ is nonconstant) there are solutions which have

isoperimetric ratios approaching inﬁnity. Second, we prove (in Sections 5

and 6) that if the hypersurface has bounded isoperimetric ratio, and α ≤

1/n, then a short time later the moving hypersurfaces are strictly convex

and have bounded curvature. The exponent 1/n is again sharp, as there are

solutions of the Gauss curvature ﬂow for any α > 1/n which remain non-

strictly convex and are not C

∞

— in fact any initial convex hypersurface

which includes a planar piece will behave this way. This phenomenon was

ﬁrst noted by Richard Hamilton for the case α = 1 [Ha1]. We describe such

behaviour more fully in Section 12 of this paper.

By combining these results, we obtain the following:

Theorem 1. For any open bounded convex region Ω

, any smooth positive

function ρ : S

→ R, and any α ∈ (1/(n + 2), 1/n], there exists a family of

embeddings x : S

× [0, T ) → R

n+1

satisfying (6), unique up to composition

with an arbitrary time-independent diﬀeomorphism, such that M

= x(S

, t)

converges in Hausdorﬀ distance to the boundary of the region Ω

as t ap-

proaches zero. x is smooth and strictly convex for t > 0 and converges to a

point p ∈ R

n+1

as t approaches T . Furthermore, the hypersurfaces



Vol(S

)

Vol(M

)



1/(n+1)

− p)

converge in C

∞

as t approaches T , to a smooth, strictly convex limit hyper-

surface

for which hx, νi = cρ(ν)K

for some c > 0.

This m eans that the evolving hypersurface contracts to a point, and

asymptotically approaches a solution which evolves purely by homotheti-

cally scaling about this limiting point.

We also have the following generalisation for smaller α:

Theorem 2. The result of Theorem 1 also holds for a solution of (6) with

any α ∈ (0, 1/n], provided the isoperimetric ratios of the evolving hypersur-

faces remain bounded.

We deduce in Section 7 the ﬁrst part of Theorem 1, that solutions contract

to points in ﬁnite time (in fact we prove this for any α > 0). This was proved

previously for isotropic cases by Tso [Ts] and Chow [Ch1]. In Section 8 we

prove that s olutions exist starting from singular initial hypersurfaces, and

immediately become smooth and strictly convex if α ≤ 1/n — note that we

make no regularity assumptions about the initial hypersurface, other than

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 7

those implied by its convexity. We also prove in Theorem 15 the existence of

unique viscosity solutions for α > 1/n, although this is not required for the

proof of Theorems 1 and 2. In Section 9 we digress from the main argument

of the pap er to apply the regularity and convexity estimates in a simple

new proof of the convergence theorem for the aﬃne normal ﬂow. In Section

11 of the paper we use Theorem 2 to deduce the existence of non-spherical

homothetic solutions of Equation (6) for constant ρ and suitable α between

0 and 1/(n + 2).

We remark that Urbas [U2] has considered noncompact solutions of iso-

tropic equations of the form (6), in particular proving the e xistence of solu-

tions which evolve by homothetically expanding or translating.

3. Notation and preliminaries.

The inradius r

−

of an open convex region is the supremum of the radii

of all balls contained in it, and the circumradius r

is the inﬁnum of the

radii of balls containing it. In this paper we refer to the ratio r

−

as the

isoperimetric ratio of the body.

For a convex region with boundary given by a smooth embedding x : M →

n+1

, we have an outward unit normal vector ﬁeld ν : M → S

⊂ R

n+1

which we use to deﬁne the Weingarten map W

: T

M → T

M by the

formula

W(u) = D

ν ∈ T

ν(x)

' T

for any x ∈ M and u ∈ T

M. The eigenvalues λ

, . . . , λ

of W(x) are the

principal curvatures of M at x. The elementary symmetric functions E

these are deﬁned by

(7) E





1≤i

<···<i

≤n

k=1

In particular, K = E

is the Gauss curvature, and H = E

is the mean

curvature.

The covariant derivative ∇ on the hypersurface is given by the formula

∇

v = D

v + hW(u), viν.

We will ﬁnd it convenient at some points in this paper to describe an

open convex region Ω ⊂ R

n+1

in terms of its support function h : S

→ R,

deﬁned by

(8) h(z) = sup

x∈Ω

hx, zi.

The support function completely describes the region Ω — in particular, Ω

can be recovered from h via the expression

Ω = ∩

z∈S



y ∈ R

n+1

: hz, yi < h(z)



8 BEN ANDREWS

Alternatively, in the case where M is strictly convex and smooth, the support

function can be used to deﬁne a canonical embedding ¯x of S

with image

equal to M:

(9) ¯x(z) = h(z)z +

∇

h(z)¯g

∇

This has the property that the outward normalto M at the point ¯x(z) is

equal to z, for each z ∈ S

The Weingarten map can also be recovered directly from h:

−1

(u), vi =

∇

h + hu, vih

where

∇ is the covariant derivative on S

, and we identify T

M and T

ν(x)

For convenience we will denote by r

the corresponding symmetric bilinear

form, the eigenvalues of which are the principal radii of curvature r

= λ

−1

i = 1, . . . , n:

(10) r

∇

h + ¯g

satisﬁes a Codazzi-type identity:

∇

h + ¯g

∇

h(11)

∇

h +



¯g

∇

h − ¯g

∇



+ ¯g

∇

Diﬀerentiating (11), commuting derivatives, and applying (11) again to the

result, we obtain a version of the Simons’ identity for the second derivatives

of the second fundamental form [Si]:

(12)

∇

+ ¯g

− ¯g

where the brackets denote symmetrisation.

For convenience, we will denote by S

the kth eleme ntary symmetric

function of the eigenvalues of r

. In particular, S

= K

−1

. S

may be

considered as a function of the components of the matrix r

, and we denote

and

ijpq

the ﬁrst and second derivatives:

∂S

∂r

and

ijpq

∂

∂r

is a positive deﬁnite symmetric bilinear form provided r is positive deﬁ-

nite, and S

1/k

is a concave function of the components of r for k = 1, . . . , n

[Mi].

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 9

The support function allows the degenerate parabolic system of equation

(6) to be re-written as a parabolic scalar equation (see [U1], [A1], [A4]):

(13)

dh(z)

= −ρ(z) det



∇

h + Idh



−α

In particular, this implies the existence of a smooth solution of Equation (6)

for a short time for any smooth, strictly convex initial hypersurface.

In a region where a family of hypersurfaces moving under Equation (6) can

be represented as graphs x

n+1

= u

, . . . , x

) for some convex functions u

we can work with an equivalent scalar parab olic equation for the functions

(14)

∂

∂t

u = ˜ρ(Du)



det D



(1 + |Du|

)

α(n+2)−1

where

˜ρ(Du) = ρ

i=1

u − e

n+1

1 + |Du|

We note some elementary features of Eq. (6): First, the speed of motion

is given by a homogeneous function of the curvatures, and this homogeneity

leads to a scaling property of solutions. Spe ciﬁcally, if x : M ×[0, T ] → R

n+1

satisﬁes Eq. (6), then for each λ > 0 another solution x

: M ×[0, λ

1+nα

T ] →

n+1

is given by

(15) x

(p, t) = λx(p, λ

−(1+nα)

t).

This also implies corresponding scaling invariance properties for the solu-

tions of Equations (13) and (14).

A second important property of solutions of Eq. (6) is the comparison

principle: If {M

(i)

}, i = 1, 2 are two families of smooth, strictly convex hy-

persurfaces moving under Eq. (6), and M

(1)

∩M

(2)

= ∅, then M

(1)

∩M

(2)

= ∅

for all t > 0 in the common interval of existence. A local version also holds:

If {M

(i)

}, i = 1, 2 are families of smooth, strictly convex hypersurfaces with

boundary, M

(1)

∩ M

(2)

= ∅, and M

(1)

∩ ∂M

(2) = M

(2)

∩ ∂M

(1) = ∅ for

t ∈ [0, T ], then M

(1)

∩ M

(2)

= ∅ for t ∈ [0, T ].

4. Monotone quantities and diameter bounds.

In this section we prove that whenever α > 1/(n + 2), the evolving hyper-

surfaces have bounded isoperimetric ratios for as long as the solution exists.

The main to ol used here is an integral estimate known as the entropy esti-

mate, which was proved for the case α = 1 by Chow [Ch1], and for other α

by the author [A3].

10 BEN ANDREWS

We deﬁne an integral quantity Z

ρ,α

for any given α and ρ by

ρ,α

= Vol(M)

n/(n+1)



ρ dµ

ρK



1/(α−1)

if α 6= 1, and

ρ,1

= Vol(M)

n/(n+1)

exp



ρ dµ

ρK log K dH



if α = 1. For convenience, we also denote by Z

\,α

the same quantity with

ρ ≡ 1.

Theorem 3. For any smooth, strictly convex solution of Equation (6),

ρ,α

≤ 0

with equality if and only if the equation hx, νi = cρ(ν)K

holds for some

c > 0 and some choice of origin in R

n+1

This integral bound will be combined with the following estimate to de-

duce isoperimetric ratio bounds for solutions of the ﬂow:

Theorem 4. For any smooth, strictly convex hypersurface M

in R

n+1

(M)

−

(M)

≤ C(α, ρ)Z

(n+1)β(α)

ρ,α

for some positive constant β(α), provided α > 1/(n + 2).

Proof. We begin with a bound in terms of Z

\,α

Following [Ha2], we begin by obtaining a lower bound on the n-dimen-

sional areas of projections of M onto hyperplanes: Given a direction z

∈ S

the area of the projection on to the plane with normal z

is given by

|hz, z

i|S

dµ.

We apply the H¨older inequality to bound this from below, as follows:

|hz, z

i|S

dµ

≥



1−α

dµ



1/(1−α)



|hz, z

1−1/α

dµ



α/(α−1)

provided α ∈ (0, 1). The integral

|hz, z

dµ is bounded for β > −1.

Hence for α ∈ (1/2, 1) we have

≥ C



1−α

dµ



1/(1−α)

= CV

n/(n+1)

−1

\,α

For α ≥ 1 this inequality still holds, because by the H¨older inequality

\,α

is increasing in α.

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 11

Finally, we consider the case where α ∈ (1/(n + 2), 1/2]: The H¨older

inequality gives

\,α

≥ Z

(n+1)(3/4−α)

4(1−α)(3n+2)

\,1/(n+2)

α−1/(n+2)

(1−α)(3−4/(n+2))

\,3/4

The aﬃne isoperimetric inequality (see [B], §26 and §73, [Sa] or [A4], The-

orem 7.1) implies that Z

\,1/(n+2)

≥ 1. Hence by applying the bound on A

in terms of Z

\,3/4

, which we know from the cases treated above, we have

≥ CV

n/(n+1)

−1

\,3/4

≥ CV

n/(n+1)

−

(1−α)(3−4/(n+2))

α−1/(n+2)

\,α

Hence for each α ∈ (1/(n+2), ∞) we have A

≥ CZ

−β(α)

\,α

for some constant

C and some positive exponent β(α).

Next we deduce a bound on the maximum width of M (the largest dis-

tance between parallel supporting hyperp lanes ): Let z

be the normal di-

rection of a pair of parallel supporting hyperplanes for M at maximal sep-

aration. T hen the points of contact of M with these two planes are joined

by a segment with length equal to the maximum width of M, and which

is entirely contained in M. Choosing the origin to be at the centre of this

segment, we have h(z

) = h(−z

) = w

/2 where w

is the maximum width

of M, and h(z) ≥ |hz, z

i|h(z

) for all z ∈ S

. But then the enclos ed volume

of M is computed by:

Vol(M) =

n + 1

dµ ≥

n + 1

h(z

)|hz, z

i|S

dµ =

2(n + 1)

Hence w

≤ 2(n + 1)Vol(M)/A

≤ CZ

β(α)

\,α

Vol(M)

1/(n+1)

Note that Vol(M) ≤ w

−

where w

−

is the minimum width of M, since

M in contained between (n + 1) pairs of parallel planes in any set of or-

thonormal directions; and, in particular, in the case where one of the pairs

of planes is at minimal separation. Then the separation of all the other pairs

is bounded by w

. It follows that w

−

≥ V w

−n

≥ CZ

−nβ(α)

\,α

Vol(M)

1/(n+1)

This gives a bound on the ratio of the minimum and maximum widths of

the hypersurface, and this is suﬃcient to bound the isoperimetric ratio (see

for example [A1], Lemma 5.4).

Finally, we consider the anisotropic cases ρ 6= const.: For α < 1, Z

ρ,α

comparable to Z

\,α

inf

1/(1−α)

\,α

≤ Z

ρ,α

≤

sup

1/(1−α)

\,α

For α ≥ 1 the desired inequality res ults from the monotonicity of Z

ρ,α

as a

function of α, a consequence of the H¨older inequality. 

12 BEN ANDREWS

We have shown in particular that for α > 1/(n +2), any solution of Equa-

tion (6) with smooth, strictly convex initial data has uniformly bounded

isoperimetric ratio on the entire interval of its existence.

5. Displacement and speed bounds.

In this section we prove that the ratio of the maximum and minimum values

of the speed remains uniformly bounded for as long as the solution exists.

We will ﬁrst deduce upper bounds on the displacement of the hypersur-

faces, by using spheres enclosed within M

as barriers:

Theorem 5. For any α > 0 and smooth positive ρ, and any smooth, strictly

convex solution {M

}

t>0

of Eq. (13),

h(z, t) ≥ h(z, 0) − C

)

−

)

1+nα

for all t ∈ (0, C

−

)

1+nα

] in the interval of existence of the solution,

where C and C

depend only on α and ρ.

Proof. Choose the origin at the centre of a ball of radius r

−

) enclosed

by M

. Fix z ∈ S

, and deﬁne for each ε ∈ (0, 1] a sphere

= {y ∈ R

n+1

: |y − (1 − ε)¯x(z)| = εr

−

)}.

Then S

is contained in the convex hull of ¯x(z) and B

−

)

(0), so by con-

vexity is contained in M

Any family of spheres of the form S

r(t)

(p) with p ∈ R

n+1

and

r(t) =



r(0)

1+nα

− sup ρ(1 + nα)t



1/(1+nα)

satisﬁes h˙x, νi ≤ −ρK

, and hence act as barriers for solutions of Eq. (6),

by the comparison principle.

This gives an estimate on the support function of M

in direction z: S

produces a barrier which shrinks to its centre at time ε

1+nα

−

(M)

1+nα

/((1+

nα) sup ρ), and we have

h(z, t) − h(z, 0) ≥ −

((1 + nα) sup ρt)

1+nα

−

(M)

h(z, 0) ≥ −C(α, ρ)

)

−

)

1+nα



Our next estimate is a speed bound, which we prove using the maximum

principle applied to the evolution equations for the spee d and the support

function. The proof is related to that given in [Ts] for the isotropic case of

Eq. (1).

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 13

Theorem 6. For any smooth, strictly convex solution {M

}

[0,T ]

of Eq. (13)

with R

−

≤ r

−

) ≤ r

) ≤ R

for t ∈ [0, T ],

ρ(z)S

−α

≤ C(n, α, ρ)

−nα

−



−



nα

1+nα

Proof. From the deﬁnition (10) and the evolution Equation (13) we obtain

(16)

∂

∂t

= −



∇

(ρS

−α

) + ¯g

ρS

−α



Since S

= det r

, this implies

(17)

∂

∂t



ρS

−α



= αρS

−(1+α)



∇

(ρS

−α

) + ¯g

ρS

−α



We also have

∂

∂t

h(z, t) = −ρS

−α

(18)

= αρS

−(1+α)



∇

h + ¯g



− (1 + nα)ρS

−α

Combining Eqs. (17) and (18), we obtain for q =

ρS

−α

h−R

−

∂

∂t

q = αρS

−(1+α)

∇

q +

2αρS

−(1+α)

h − R

−

∇

− q

(αR

−

H/2 − (1 + nα))

where H =

i=1

−1

= nS

n−1

≥ nS

−1/n

. By the maximum principle,

this implies the following inequality for Q = sup

≤ −Q



C(n, α, ρ)R

−

nα

− (1 + nα)



and we deduce

Q ≤ max



C(α)

nα

−

, C

(α)R

−

nα

1+nα

−

1+nα

−

nα

1+nα



From the deﬁnition of Q and the estimate h ≤ 2R

, we have

ρS

−α

≤ max

(

C(n, α, ρ)R

−nα

−

, C

(n, α, ρ)



−



nα

1+nα

−

nα

1+nα

)



We now proceed to obtain lower bounds on the speed and displacement.

It is in this estimate that we require α ≤ 1/n. The argument combines

barrier arguments with a Harnack inequality (proved for isotropic Gauss

curvature ﬂows by Chow [Ch3] and for more general ﬂows by the author

[A7]).

14 BEN ANDREWS

The Harnack estimate can be stated as follows for any solution of Equation

(13) (see [A7], Theorem 5.6):

Theorem 7. For any smooth, strictly convex solution of (13) on S

×[0, T ),



ρS

−α

nα

nα+1



≥ 0

everywhere on S

× (0, T ).

Our lower speed estimate is the following:

Theorem 8. For α < 1/n the following holds for any smooth, strictly con-

vex solution of Eq. (13):

h(z, t) ≤ h(z, 0) − C(ρ, n, α)r

)

−

2nα

1−nα

and

ρ(z)S

(z, t)

−α

≥ C

(ρ, n, α)r

)

−

2nα

1−nα

nα

1−nα

for 0 < t < C

(n, ρ, α)r

)

1+nα

. For α = 1/n we have instead the

estimates for each γ > 0

h(z, t) ≤ h(z, 0) − C(n, ρ, γ)r

)

1+2γ

−γ

−C

(ρ,n,γ)r

)

−n

and

ρ(z)S

(z, t)

−1/n

≥

C(n, ρ, γ)r

)

2γ−1

−γ

−

(ρ,n,γ)r

)

−n

for 0 < t <

(n, ρ, γ)r

)

Proof. For n = 1 these estimates are proved in [A2], Theorem II2.4. Suppose

n ≥ 2.

In the case α < 1/n it suﬃces to use large spheres as barriers: Fix

z ∈ S

. Then M

is enclosed in a hemispherical region obtained by in-

tersecting the sphere of radius 2r

) centred at ¯x(z) with the half-space

{y ∈ R

n+1

: hy, zi ≤ h(z)}. For any ε < 2r

(M) this hemispherical region is

enclosed by the sphere S

of radius (ε

+4r

(M)

)/(2ε) centred at the point

¯x(z)−(4r

(M)

−ε

)/(2ε)z (this sphere is chosen to have support function in

direction z equal to h(z) + ε). We consider the evolution of these spheres for

suitably small ε: Since ρ ≥ inf

ρ, a sphere of radius r evolves in time t to be

contained inside a sphere of radius



1+nα

− (1 + nα) inf ρt



1/(1+nα)

about

the same centre, which is enclosed by the sphere of radius r − inf ρr

−nα

In particular, this applies for each of the sphere S

, and by the compari-

son principle M

is also enclosed by this smaller sphere. This implies the

inequality

h(z, t) − h(z, 0) ≤ ε − (inf ρ)



+ 4r

(M)

2ε



−nα

≤ ε



1 − inf ρ(4r

(M)

)

−nα

nα−1



MOTION OF HYPERSURFACES BY GAUSS CURVATURE 15

In particular, cho osing

ε =



t inf ρ

1+2nα

)

2nα



1/(1−nα)

we obtain

h(z, t) − h(z, 0) ≤ −



inf ρ

1+2nα

)

2nα



1/(1−nα)

for t ≤ C(ρ, n, α)r

)

1+nα

This estimate on the change in the support function can be converted

to an estimate on the speed using the Harnack estimate from Theorem 7:

Applying the estimate on the time interval [t/2, t], we have

)

−

2nα

1+nα

1−nα

≤ h(z, t/2) − h(z, t)

= ρ(z)

t/2

(z, τ )

−α

dτ

≤ t/2 sup

τ∈[t/2,t]



ρS

−α



Theorem 7 then gives

nα

nα+1

ρ(z)S

(z, t)

−α

≥ (t/2)

nα

nα+1

sup

[t/2,t]

ρS

−α

and hence

ρ(z)S

(z, t)

−α

≥ C

)

−

2nα

1+nα

1/(1−nα)−1

In the case 1/n the sphere barriers are not suﬃcient, and we instead work

with graphical barriers. The displacement bound is a consequence of the

following:

Lemma 9. Suppose α = 1/n. If M

has bounded isoperimetric ratio, and

lies in the region x

n+1

≥ 0 and within the ball kxk ≤ R, then M

lies inside

the region

n+1

≥ C

1+2γ

−γ



−C

2n−2

(r−2R)

−n

+ e

−C

2n−2

(r+2R)

−n



for 0 ≤ t ≤ C

, where r

i=1

, and C

, C

and C

are constants

depending only on n, ρ, and γ.

Proof. We will show that the boundary of the region described is a graphical

subsolution of the evolution Equation (14). In the special case of a radially

symmetric function, we have

K =

)

n−1

(1 + (u

)

(n+2)/2

16 BEN ANDREWS

and so

(19)

˙u − ˜ρ(Du)K

1/n

1 + |Du|

≤ ˙u − inf ρ(u

)

1/n





1−1/n



1 + (u

)



−1/n

A direct computation shows that the function

u(r, t) = C

1−2γ



−C

2n−2

(r−2R)

−n

+ e

−C

2n−2

(r+2R)

−n



makes the right-hand side of (19) non-positive on the region r < R, t ≤

, for any γ ≥ 0, where C

, C

, and C

depend on n, ρ and γ. Since the

boundary of this region cannot intersect the hypersurface M

, the compari-

son principle applies. 

This gives the bounds in the theorem, since we can rotate and translate

the solution to bring the initial supporting hyperplane to the hyperplane

n+1

= 0, with M

satisfying the conditions of Lemma 9 with R = 2r

Thus h(z, 0) = 0. For positive suﬃciently small t, Lemma 9 gives

h(z, t) = −inf

n+1

≤ −u(0, t)

as required.

Similar barriers can also be constructed for each α < 1/n.

The speed bound follows using Theorem 7 as for the previous cases.

We remark that the estimate for α < 1/n does not rely at all on the

particular structure of the Gauss curvature ﬂows — the same result holds

for any strictly parabolic ﬂow with speed homogeneous of degree less than

1 in the curvatures. 

6. Curvature control.

In this section we prove that the ratio of the maximum and minimum princi-

pal curvatures remains bounded throughout the evolution, given the upper

and lower speed bounds of the previous section. Our argument is an appli-

cation of the parabolic maximum principle to the evolution equation for the

curvature.

In the case n = 1 the speed bounds above and below already give complete

control on the curvatures. For the rest of this section we assume n ≥ 2.

Theorem 10. Suppose h : S

× [0, T ] is a solution of Eq. (13) for which

the isoperimetric ratio is bounded and the speed is bounded above and below

— that is, there exist constants C

, C

such that

0 < C

≤ S

(z, t) ≤ C

for every z ∈ S

and t ∈ [0, T ]. Then there exist positive const ants C

and

depending only on C

, C

, ρ, n and α such that

(z, t) ≥ min{C

n−1

, C

}

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 17

for all i ∈ {1, . . . , n}, z ∈ S

, and t ∈ [0, T ].

Proof. We begin by computing the evolution equation for the matrix r

∇

h + ¯g

h under Eq. (13):

∂

∂t

= −

∇



ρS

−α



− ¯g

ρS

−α

= αρS

−(1+α)

∇

− α(1 + α)ρS

−(2+α)

∇

+ αρS

−(2+α)

klmn

∇

− ¯g

ρS

−α

− S

−α

∇

+ αS

−(1+α)

∇

+ αS

−(1+α)

∇

In the ﬁrst term here we apply the identity (12), to yield:

∂

∂t

= αρS

−(1+α)

∇

− α(1 + α)ρS

−(2+α)

∇

(20)

+ αρS

−(2+α)

klmn

∇

+ (nα − 1)ρS

−α

¯g

− αρS

−(1+α)

¯g

+ S

−α

∇

ρ + αS

−(1+α)

∇

+ αS

−(1+α)

∇

We wish to obtain an upper bound for the eigenvalues of r

, so the second

term on the ﬁrst line and the last term on the second line are good terms

since they are negative. The ﬁrst term of the ﬁrst line is an elliptic operator,

and so is non-positive at a point and direction where a maximum eigenvalue

occurs. The ﬁrst term of the second line we estimate using the concavity

of the nth root of the determinant as a function of the components of the

matrix, which is equivalent to the inequality

(21)



klmn

−

n − 1



≤ 0

for any symmetric matrix ξ. Finally, the ﬁrst term on the last line is

bounded, and the other two terms of the last line can be estimated in terms

of the good second term of the ﬁrst line:



∇



≤ Cε



∇



+ Cε

−1

for any ε > 0. Combining these estimates, we obtain

(22)

∂

∂t

≤ αρS

−(1+α)

∇

+ CS

−α

¯g

− αρS

−(1+α)

¯g

The last term here will allow us to obtain an estimate independent of

initial data: We have S

¯g

= S

n−1

, and the Newton inequalities [Mi]

give

(23) S

n−1

≥ S

n−2

n−1

≥ CS

n−2

n−1

max

18 BEN ANDREWS

Now work at a p oint and time where a maximum eigenvalue is attained,

and suppose i = j and e

is the eigenvector of r with the largest eigenvalue.

Then the ﬁrst term in the evolution equation is negative, and

∂r

max

∂t

≤ CS

−α

¯g

− CS

−(

n−1

+α)

n−1

max

≤ CS

−α



¯g

− CS

−

n−1

max



Given the bound below on S

−α

, the bracket is negative provided r

max

suﬃciently large. For the same reason the coeﬃcient in front of the bracket

does not become small, and we have for r

max

suﬃciently large

max

≤ −Cr

n/(n−1)

max

The result now follows by the parabolic maximum principle and comparison

with the solution of the ordinary diﬀerential equation du/dt = −Cu

n/(n−1)



Next we observe that this automatically provides an upper bound on the

principal curvatures:

Proposition 11. Let W be a positive deﬁnite symmetric matrix for which

W ≥ εId and detW ≤ C. Then W ≤ Cε

−(n−1)

Id.

Proof. Number the eigenvalues λ

of W in ascending order: λ

≤ λ

≤ ··· ≤

. Then

. . . λ

n−1

≤

n−1

where K = λ

. . . λ

= det W. 

In particular, the upper speed bound of Theorem 6 and the lower curva-

ture bound of Theorem 10 imply an upper curvature bound:

Corollary 12. Un der the conditions of Theorem 10 there exist constants

and C

such that

W ≤ max{C

−(n−1)

, C

}Id.

7. Convergence to a point.

In this section we prove that any solution of Eq. (6) with a smooth, strictly

convex initial hypersurface converges to a point in ﬁnite time. In the special

case of isotropic ﬂows (ρ ≡ 1) this was proved by K.S. Chou [Ts] for α = 1

and by Ben Chow [Ch1] for other α. While we only need the result for

α ≤ 1/n, we give a proof which works for larger α as well.

Theorem 13. For any α > 0 and positive ρ ∈ C

∞

), and any smooth,

strictly convex hypersurface M

⊂ R

n+1

, the h ypersurfaces M

given by the

solution of Eq. (6) exist for a ﬁnite time T and converge in Hausdorﬀ dis-

tance to p ∈ R

n+1

as t approaches T .

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 19

Proof. The maximal time of existence must be ﬁnite: By the comparison

principle, if M

is enclosed by a sphere S

r(0)

(q) for some r > 0 and q ∈ R

n+1

then for all t in the interval of existence, M

is enclosed by the sphere S

r(t)

(q),

where

r(t) =



r(0)

1+nα

− (inf

ρ)(1 + nα)t



1/(1+nα)

r(t) converges to zero in ﬁnite time, and M

cannot exist beyond this time.

Consider again the estimate (22) for the evolution of the curvature. We

also have the evolution equation

∂

∂t

h = −ρS

−α

(24)

= αρS

−(1+α)

∇

h + αρS

−(1+α)

¯g

h − (1 + nα)ρS

−α

Combining these, we obtain

∂

∂t

+ Ah¯g

) ≤ αρS

−(1+α)

∇

+ Ah¯g

)

+ (C − A(1 + nα))S

−α

¯g

+ (Ah − r

) αρS

−(1+α)

¯g

Choose A = C/(1 + nα), so that the last term of the ﬁrst line vanishes.

Also note that since the hypersurfaces are contracting, we have h ≤ h

sup

h(z, 0) as long as the solution exists. Therefore we have, writing q

+ Ah¯g

∂

∂t

≤ αρS

−(1+α)

∇

− αρS

−(1+α)

¯g

− 2h

¯g

)

and hence by the parabolic maximum, the maximum eigenvalue of q

decreasing if it is larger than 2h

. Since the initial hypersurface is smooth

and strictly convex, q

is bounded at t = 0. Therefore we have a uniform

bound on q

and hence r

throughout the interval of e xistence.

Suppose the inradius of the hypersurfaces M

do not converge to zero —

that is, the solution exists for a maximal time interval [0, T ), but there is

some ball of positive radius that remains enclosed by the solution through-

out. By the argument in Section 5, the speed remains bounded throughout

the interval of existence.

By Proposition 11 this also implies a bound on the curvature, so that

the hypersurfaces remain uniformly smooth and strictly convex on the time

interval [0, T ). It follows that there exists a subsequence of times {t

} con-

verging to T such that M

converges in C

∞

to a smooth, strictly convex

limit M

. Furthermore, the C

∞

convergence implies that all time deriva-

tives converge, so that in fact M

approaches M

in C

∞

as t approaches

20 BEN ANDREWS

T . Hence we have a smooth solution on [0, T ], and the short-time exis-

tence result implies that this can be extended beyond T , contradicting the

assumption that T was maximal.

Therefore the inradius converges to zero. Since r

is uniformly bounded,

this also implies that the circumradius converges to zero, and the hypersur-

faces converge to a point. 

Theorem 13 implies a bound below on the time of existence of solutions

with smooth and strictly convex initial data, in terms of the inradius of the

initial hypersurface, ρ, and α: Any sphere which is initially enclosed by the

hypersurface acts as a barrier, preventing the hypersurface from contracting

to a point too quickly.

8. Short-time existence.

In this section we prove the following:

Theorem 14. For any positive ρ ∈ C

∞

), α ≤ 1/n, and open bounded

convex region Ω

⊂ R

n+1

, there exists a smooth, strictly convex solution

: S

× (0, T ) of Eq. (6) which converges to M

= ∂Ω

in Hausdorﬀ

distance as t approaches zero. Any other such solution y

: S

× (0, T

) is

given by x

◦ ϕ for some smooth diﬀeomorphism ϕ of S

At the end of the section we also prove the existence and uniqueness of

viscosity solutions for arbitrary convex initial data and arbitrary α > 0.

8.1. Existence. In order to construct a solution which approaches M

the initial time, we consider a family of smooth, strictly convex hypersurfaces

(ε)

which approach M

in Hausdorﬀ distance as ε approaches zero. By

Theorem 13, for each ε > 0 there exists a unique solution M

(ε)

of Eq. (6)

with initial condition M

(ε)

, which converges to a point in ﬁnite time T

> 0.

Then we have

, M

(ε)

) ≤ d

, M

(ε)

) + d

(ε)

, M

(ε)

)

≤ d

, M

(ε)

) + Ct

1/(1+nα)

By Theorem 10 and Corollary 12, the hypersurfaces M

(ε)

satisfy bounds

above and below on the principal curvatures, uniformly in ε over every

compact subset of S

× (0, T ). It follows from the regularity theory for

solutions of uniformly parabolic equations concave in the second derivatives

([K], Theorem 5.5) that there are also bounds on all higher derivatives of the

curvatures, uniformly in ε over every compact subset of S

×(0, T ). It follows

from the Arzela-Ascoli theorem that there exists a sequence ε

approaching

zero such that {M

(ε

)

} converges in C

∞

to a family of hypersurfaces {M

}

satisfying the same bounds. In particular, {M

} satisﬁes Eq. (6) on S

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 21

(0, T ), M

is smo oth and strictly convex for each t > 0, and d

, M

) ≤

1/(1+nα)

8.2. Uniqueness. Suppose we have two solutions {M

(1)

} and {M

(2)

} of

Eq. (6), both converging to M

in Hausdorﬀ distance as t → 0, and denote

by h

(i)

the corresponding support functions. Fix ε > 0. Then there exists

(ε) > 0 such that |h

(i)

(z) − h

(z)| < ε for i = 1, 2 and all z ∈ S

and

t ∈ (0, t

(ε)). Choose a smooth, strictly convex hypersurface M

(ε)

with

support function h

(ε)

such that h

(z) − 2ε < h

(ε)

(z) < h

(z) − ε. Without

loss of generality we assume that the origin is at the centre of a ball of radius

−

) enclosed by M

. Then r

−

) ≤ h

(z) ≤ 2r

). It follows that

(ε)

(z) < h

(z) −ε < h

(i)

(z) < h

(z) + ε <



1 +

3ε

−

) − 2ε



(ε)

for i = 1, 2 and all z ∈ S

and t ∈ (0, t

(ε)). The comparison principle and

the scaling property given by Eq. (15) then imply

(ε)

(z) < h

(i)

t+τ

(z) < (1 + λ)h

(ε)

(1+λ)

−(1+nα)

(z)

for all τ ≥ 0 for which these all exist, where λ =

3ε

−

)−2ε

. Consequently,



(2)

t+τ

(z) −h

(1)

t+τ

(z)



≤ (1 + λ)h

(ε)

(1+λ)

−(1+nα)

(z) −h

(ε)

(z)

≤ λh

(ε)

(1+λ)

−(1+nα)

(z) +



(ε)

(1+λ)

−(1+nα)

(z) −h

(ε)

(z)



≤ 2r

)λ + C





1 − (1 + λ)

−(1+nα)





1/(1+nα)

≤ Cλ + C(λτ)

1/(1+nα)

Here we used Theorem 6 to obtain the second-last line. Now take t → 0.

Since ε > 0 is arbitrary and C independent of ε, we have for each τ > 0 and

z ∈ S

(2)

(z) = h

(1)

(z).

Note that the proof presented here does not rely strongly on the particular

structure of the evolution Equation (6). In particular, the uniqueness argu-

ment is valid for any ﬂow by a monotone, positively homogeneous function

of curvature, since the bound on the change in the support function given in

Theorem 6 also holds for all such evolution equations. The existence argu-

ment requires a speed bound and regularity estimates independent of initial

data.

We now proceed to the case α > 1/n: In this case (as we show in Section

12) one c annot expect to produce smooth solutions from arbitrary convex

initial hypersurfaces. Instead we will work with a weaker notion of solution:

A family of convex regions {Ω

}

0<t<T

is called a viscosity solution of Eq. (6)

22 BEN ANDREWS

if the following conditions hold: First, for any smooth, strictly convex hy-

persurface M

contained in Ω

for some t

∈ (0, T ), the hypersurfaces M

given by solving (6) are contained in Ω

for all t ∈ [0, T −t

) in the domain

of existence of the M

. Second, for any smooth, strictly convex hypersurface

which encloses Ω

for some t

∈ (0, T ), the hypersurfaces M

enclose

Ω

for all t ∈ [0, T − t

Theorem 15. For any smooth positive ρ ∈ C

∞

), α > 0, and any open

bounded convex region Ω

⊂ R

n+1

, there exists a unique viscosity solution

{Ω

}

0<t<T

which converges to Ω

is Hausdorﬀ distance as t approaches zero.

Ω

converges to a point as t approaches T .

Proof. We use the same construction as presented in the proof of Theorem

14, producing a solution {M

(ε)

} for each ε > 0, with M

(ε)

approaching ∂Ω

in Hausdorﬀ distance as ε approaches zero. We specify further that M

(ε)

contained in Ω

for all ε > 0, and is increasing in ε.

For ε suﬃciently small, we can choose an origin for R

n+1

and radii R >

r > 0 such that the ball B

(0) is enclosed by all of the hypersurfaces M

(ε)

and the ball B

(0) contains all of the hypersurfaces M

(ε)

. By the comparison

principle, there exists δ > 0 such that the ball B

r/2

(0) is enclosed by all the

hypersurfaces M

(ε)

for t ∈ [0, δ]. The hypersurfaces also remain enclosed by

the ball B

(0).

It follows that the support functions h

(ε)

(z) are uniformly Lipschitz: By

the Formula (9), we have |¯x|

= h

+ |

∇h|

, and re-arrangement gives

∇h|

≤ |¯x|

≤ R

, which is a uniform Lipschitz bound.

Furthermore, the displacement bound and the speed bound of Theorem

6 show that h

(ε)

is H¨older continuous in t, uniformly in ε and z, and also

uniformly Lipschitz on compact subsets of (0, δ). Therefore h

(ε)

(z, t) is a

H¨older continuous function on S

× [0, δ], uniformly in ε. By the Arzela-

Ascoli theorem, there exists a sequence ε

approaching z ero which converges

to a limit h(z, t) satisfying the same e stimate s. By the Blaschke selection

theorem, each of the functions h

= h(., t) is the support function of a convex

region Ω

, and the same argument as in the proof of Theorem 14 shows that

Ω

approaches Ω

in Hausdorﬀ distance as t approaches zero.

We need to prove that the family {Ω

} is a viscosity solution. The ﬁrst

condition is easily checked: If M

is contained within Ω

, then M

is also

enclosed by M

(ε)

for ε > 0 suﬃciently small. By the comparison principle,

the resulting solution M

is enclosed by M

(ε)

for t > 0, and also M

(ε)

increasing in ε and converges to ∂Ω

as ε approaches zero. Therefore M

contained in Ω

for t > 0.

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 23

The second condition also follows easily: Any hypersurface M

which

encloses Ω

also encloses all of the hypersurfaces M

(ε)

, and so by the com-

parison principle M

encloses M

(ε)

for ε > 0 and t > 0, and so also encloses

the limit ∂Ω

The uniqueness statement in Theorem 15 follows exactly as in the proof

of uniqueness in Theorem 14, and the same argument shows that the regions

Ω

converge to a point. 

9. Application: The aﬃne normal ﬂow.

In this section we apply the speed and curvature bounds of the previous

sections to give a short proof of the following theorem:

Theorem 16. Let α = 1/(n + 2) and ρ ≡ 1. For any convex open region

Ω

⊂ R

n+1

there exists a smooth family of strictly convex embeddings x

→ R

n+1

satisfying Eq. (6) for which the Ha usdorﬀ distance between

= x

) and M

approaches zero as t → 0. Any other such solution {˜x

}

is related to {x

} by composition with a time-independ ent diﬀeomorphism.

converges to a point p ∈ R

n+1

as t approaches a ﬁnite time T , and



Vol(S

)

Vol(M

)



1/(n+1)

− p)

converges in C

∞

to an ellipsoid centred at the origin.

This theorem was proved in the case of smooth, strictly c onvex M

[A4]. The results of Section 6 allow us to give a proof which works also

for singular initial hypersurfaces. The argument is also considerably simpler

because it avoids the com plicated third-derivative estimate which was the

key to the proof in [A4]. On the other hand, we use the result that elliptic

aﬃne hyperspheres are ellipsoids, which was not necessary for the proof in

[A4].

Proof. By Section 8, we have a unique solution of Eq. (6) with the given

initial condition. Since this is smooth and strictly convex for t > 0, the

result of Theorem 13 implies that this solution converges to p ∈ R

n+1

ﬁnite time T .

In the proof we use the fact that the evolution equation is invariant under

the action of the special aﬃne group: If {M

} is a family of hypersurfaces

moving under Eq. (6), then {L(M

)} is also such a family, for any special

aﬃne transformation L.

Fix t ∈ [0, T ). There exists a special aﬃne transformation L

such that

) has

−

Vol(M

)

1/(n+1)

≤ Vol(S

)

1/(n+1)

)

(z) ≤ C

Vol(M

)

1/(n+1)

for some absolute constants C

24 BEN ANDREWS

We consider the solution

h given by the scaling relation (15):

h(z, τ ) =



Vol(S

)

Vol(M

)



1/(n+1)

z, t +



Vol(S

)

Vol(M

)



−2/(n+2)

where h

is the support function of the convex body obtained by apply-

ing the special aﬃne transformation L to the body with support func-

tion h. Then C

−

≤

h(z, 0) ≤ C

. By the comparison principle we also

have

−

≤

h(z, τ ) ≤ C

for τ ∈ [0, δ], where δ = ( C

−

)

2(n+1)/(n+2)

(1 −

−2(n+1)/(n+2)

)(n + 2)/2(n + 1). Hence on the interval [δ/2, δ] there are

uniform speed and displacement bounds (by Theorem 6), a uniform lower

bound on the speed (by Theorem 8), uniform bounds above and below on

the principal curvatures (by Theorem 10 and Corollary 12), and uniform

bounds on all higher derivatives of the curvature (by Theorem 5.5 of [K]).

It follows that the original solution satisﬁes uniform bounds on all quan-

tities which are both scaling invariant and special-aﬃne invariant, on the

time interval



t +

CVol(M

)

2/(n+2)

, t + CVol(M

)

2/(n+2)



, for some abso-

lute constant C. Since t is arbitrary, we have such bounds on the entire

interval [T /2, T ).

Therefore there exists a sequence {t

} approaching T , and a sequence of

special aﬃne transformations {L

}, such that the hyp e rsurfaces {L

(

)}

converge in C

∞

to a smooth, strictly convex limit

Suppose

does not satisfy the condition K

1/(n+2)

= chx, νi for some

c > 0 and some choice of origin. Then the time derivative of Z

\,1/(n+2)

is strictly negative , by Theorem 3. By the C

∞

convergence and scaling,

there exists k

, δ > 0, and ε > 0 such that whenever k ≥ k

we have

\,1/(n+2)



+δVol(M

)

2/(n+2)



≤ Z

\,1/(n+2)

) − ε.

But since Z

\,1/(n+2)

is non-increasing, this would imply Z

\,1/(n+2)

(

) →

−∞ as k → ∞, which is impossible. Therefore

satisﬁes the required

condition.

By Theorem 1 of [Ca], a smooth, strictly convex hypersurface satisﬁes

the condition K

1/(n+2)

= chx, νi if and only if it is an ellipsoid.

The stronger convergence statements in the Theorem follow by consid-

ering the linearization of the evolution Equation (13) about the space of

ellipsoids — a direct calculation shows that this space is strictly linearly

stable, so Proposition 9.2.3 of [Lu] and a scaling argument implies that M

converges in C

∞

to the ellipsoid

after rescaling. The details of this

argument are given for a related evolution equation in [A10], Propositions

40-41. 

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 25

Corollary 17. For any open bounded convex region Ω

⊂ R

n+1

, the follow-

ing generalised aﬃne isoperimetric inequality holds:

lim

t→0

n+2

(Ω

) ≥ 1

with equality if and only if Ω

is an ellipse.

In eﬀect this result allows a deﬁnition of the aﬃne surface area for convex

hypersurfaces which may be non-smooth or non-strictly convex, in such a

way that the aﬃne isoperimetric inequality remains true. A related exten-

sion of the aﬃne surface area has been given in [Le].

10. Proof of the main Theorems.

In this section we complete the proofs of Theorems 1 and 2. The proof is

similar to that presented in the special case of the aﬃne normal ﬂow in the

previous section, but is somewhat simpler because we do not require the

machinery of aﬃne corrections to obtain bounded isoperimetric ratios.

Section 8 provides us with a unique solution {M

} of Eq. (6) for a short

time, and Theorem 13 ensures that this solution remains smooth until it

converges to a point p ∈ R

n+1

in ﬁnite time T . In the case α > 1/(n +

2), Theorem 4 gives a uniform bound on the isoperimetric ratios of the

hypersurfaces M

throughout the interval [0, T ]. In the case described in

Theorem 2, we also have such an estimate by hypothesis.

Theorems 6, 8 and 10 and Corollary 12 therefore imply uniform bounds

above and below on the principal curvatures of the rescaled hypersurfaces

{

} deﬁned by rescaling to ﬁxed enclosed volume about the point p. The-

orem 5.5 of [K] and Schauder estimates ([Li], Theorem 4.9) imply uniform

bounds on all higher derivatives of curvature. It follows that there exists a

subsequence of times {t

} approaching T for which the hypersurfaces

converge in C

∞

to a smooth, strictly convex limit

. By the same argu-

ment as that in Section 9,

satisﬁes the equation hx, νi = cρK

for some

c > 0. The convergence in C

∞

as t approaches T follows from

Theorem 2 of [A6].

11. Application: Non-trivial homothetic solutions.

In this section we give an application of Theorem 2 to prove the existence

of homothetically contracting solutions of the isotropic ﬂows

∂

∂t

x = −K

for s uﬃciently small α. The idea is to consider the evolution of hypersurfaces

close to the sphere S

, possessing suitable symmetries. Precisely, our result

is the following:

26 BEN ANDREWS

Theorem 18. Let Γ be a proper subgroup of SO(n + 1) such that for every

z ∈ S

, the orbit of z under Γ spans R

n+1

(that is, the inclusion of Γ in

SO(n + 1) is an irreducible representation). Let λ be the smallest eigenvalue

corresponding to a non-trivial Γ-invariant spherical harmonic ϕ. Then for

α ∈ (0, 1/(λ−n)) there exists a non-spherical, Γ-symmetric, smooth, strictly

convex hypersurface satisfying the identity hx, νi = K

Proof. The linearization of the normalised isotropic Equation (6) about the

sphere solution h ≡= 1 is given by

∂

∂t

η = α(∆ + n)u + u

where ∆ is the Laplacian on S

. In particular, for α ∈ (0, 1/(λ − n)) the

h ≡ 1 solution is strictly unstable in the direction ϕ. By [Lu], Theorem

9.1.3, there exists a Γ-symmetric solution {M

} of Eq. (6) which converges

to h ≡ 1 as t → −∞ and diverges exponentially from h ≡ 1.

We observe that Γ-symmetry of a convex hypersurface implies a bound

on the isoperimetric ratio:

Lemma 19. For any Γ satisfying the conditions of Theorem 18, there exists

a const ant C such that every Γ-symmetric convex hypersurface M ⊂ R

n+1

satisﬁes r

(M)/r

−

(M) ≤ C.

Proof. If this is not the case, then we can ﬁnd a sequence of Γ-symmetric

convex hypersurfaces M

such that r

) = 1 and r

−

) ≤ 1/k. By

the Blaschke selection theorem ([Sc], Theorem 2.5.14) we can choose a sub-

sequence M

which converges in Hausdorﬀ distance to a limit M

∞

which

is again Γ-symmetric but has r

−

∞

) = 0 and r

∞

) = 1. It follows

that M

∞

is contained in a lower-dimensional subspace of R

n+1

. But this

is impossible, because there exists x ∈ M

∞

with |x| = 1; by Γ- symm etry

all of the point g(x) are in M

∞

, but these are not contained in any such

sub-space. 

Therefore we can apply Theorem 2 to the solution {M

}, obtaining C

∞

convergence to a limit M

satisfying the required identity (possibly after

scaling to ensure c = 1). M

is Γ-symm etric, and has Z strictly less than

that for the sphere solution, since Z has strictly decreased along the solution

}. Therefore M

is non-spherical. 

Corollary 20. In the case n = 1, for each k ≥ 3 and each α ∈ (0, 1/(k

−

1)) there exists a non-circular strictly convex smooth curve C

with k-fold

symmetry which contracts homothetically under the ﬂow

∂

∂t

x = −κ

where κ is the curvature, and n the unit normal.

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 27

Proof. For k ≥ 3 the subgroup Γ

generated by rotation through 2π/k satis-

ﬁes the conditions of Theorem 18. The ﬁrst spherical harmonics symmetric

under Γ

are cos(kθ) and sin(kθ), with corresponding eigenvalue λ = k

. 

Corollary 21. In the case n = 2, there exists a homothetically contracting

solution of Eq. (6) with tetrahedral symmetry provided α ∈ (0, 1/10); there

exists one with octahedral symmetry provided α ∈ (0, 1/18); and there exists

one with icosahedral symmetry provided α ∈ (0, 1/40).

Proof. In this case the only subgroups satisfying the required condition are

the s ymmetry groups of the platonic solids. There are three such groups,

since the dual solids have the same symmetry group. The ﬁrst tetrahedrally

symmetric spherical harmonic is given by the restriction of the function

xyz to S

, and the corresponding eigenvalue is 12. The ﬁrst octahedrally

symmetric spherical harmonic is x

+ y

+ z

−3x

−3y

, and

the corresponding eigenvalue is 20. Finally, the ﬁrst icosahedrally symmetric

spherical harmonic is

231z

− 315z

+ y

) + 105z

+ y

)

− 5(x

+ y

)

+ 42zx

− 420zx

+ 210zxy

and the corresponding eigenvalue is 42. 

Corollary 22. For n ≥ 3, there exists a non-spherical homothetically con-

tracting solution with the symmetry of a regular (n + 2)-simplex for α ∈

(0, 1/(2(n + 3))), and one with the symmetries of a regular hypercube for

α ∈ (0, 1/3(n + 4)).

Proof. The function

−(6/n)

i6=j

has the symmetry of a regular

hypercube in R

n+1

, and its restriction to S

is a spherical harmonic with

eigenvalue 4(n + 3).

For all n ≥ 1 there exists a cubic homogeneous harmonic polynomial u

on R

n+1

with the symmetries of a regular simplex. These are given by the

recursive deﬁnitions

, x

) = x

− 3x

n+1

, . . . , x

n+2

) = u

, . . . , x

n+1

)

+ β



n+2

−

n + 1



+ ··· + x

n+1





where β

√

2 and

k+1

= β

(k + 1)

(k + 3)

The restriction of u

to S

is a spherical harmonic with the required sym-

metries, and the corresponding eigenvalue is 3(n + 2). 

28 BEN ANDREWS

In [A5] we prove that the only homothetically contracting solutions of

ﬂows by positive powers of curvature are those given in Corollary 20. For

n = 2 we expect that there are many more solutions which are not described

by Corollary 21. In particular, for small α there should be many solutions

symmetric under each of the platonic symmetry groups, and there should

also be solutions symmetric under some other subgroups of SO(3), such as

the subgroup of rotations about a ﬁxed axis, and its discrete subgroups. In

the case n ≥ 3 there are of course many more examples of suitable subgroups

Γ which we have not mentioned explicitly in Corollary 22.

12. Hypersurfaces with planar or cyl indrical parts.

In this section we demonstrate that the result of Theorem 1 no longer holds

for any α > 1/n. Speciﬁcally, we show that any solution starting from a

hypersurface containing a planar region must still contain a planar region for

small positive times. We also consider the behaviour of hypersurfaces which

contain regions which are cylindrical or loc ally have the form M

n−k

× R

for some k > 0.

Theorem 23. Suppose M

is a compact convex hypersurface, and F

a sub-

set of M

which has the form N

n−k

×U

, where N

is a smooth convex hy-

persurface in R

n+1−k

and U

is an open subset of R

. Then for any smooth

positive function ρ on S

and any α > 1/k, the viscosity solution {M

}

starting from M

contains an open subset F

of F

for t > 0 suﬃciently

small.

Conversely, suppose Ω

in a bounded open convex set in R

n+1

for which

n−k

= inf

x∈∂Ω

sup

P,Γ



κ : y ∈ Ω

⇒ hx − y, P

⊥

i ≥

κ|π

(x − y)|



> 0

where the supremum is over all supporting hyperplanes P of Ω

which con-

tain x and all n −k-dimensional aﬃne subspaces Γ of P through x, and P

⊥

is the unit normal to P which points outward from Ω

. Then for any ρ and

any α ∈ (0, 1/k] the viscosity solution {Ω

} of (6) with initial condition Ω

is smooth a nd strictly convex for t > 0 and remains so until it contracts to

a point.

When Ω

is smooth, σ

n−k

is strictly positive if and only if E

n−k

(W) is

strictly positive, or equivalently if and only if the sum of the smallest k + 1

principal curvatures is strictly positive at every point.

Proof. To prove the ﬁrst part of the Theorem, we will construct barriers with

cylindrical symmetry, desc ribed by embeddings of the form ϕ : S

n−k

×B

(0)

with ϕ(z, x) = (u(|x|)z, x) ∈ R

n+1−k

× R

' R

n+1

. We consider the case

where u is concave and decreasing in |x|. The metric and second fundamental

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 29

form are given by

= u

¯g

= 0

= δ

|x|

)

and

1 + (u

)

= 0

= −

(1 + (u

)

3/2

|x|

−

|x|

1 + (u

)



−

|x|



The Gauss curvature is given by the expression

K = (−1)

)

k−1

n−k

|x|

k−1

(1 + (u

)

(n+2)/2

and the unit normal is

ν =

z − u

ˆx

1 + (u

)

where ˆx = x/|x|.

It follows that any function u(r, t) satisfying the inequality

(25) ˙u ≤ −sup ρ(−u

)

(−u

)

α(k−1)

−α(n−k)

|x|

−α(k−1)



1 + (u

)



1−α(n+2)

has a cylindrical graph which acts as an inner barrier for convex solutions

of Eq. (6). We proceed to construct such barriers.

Lemma 24. For given k, ρ, and α > 1/k there exist constants c

, c

and

such that for any λ > 0 and u

> 0, R > 0, the function

u = u





1 − c

(k, α)



|x|

+ c

(ρ, k, α)





2kα

1+nα

− 1



k+1−1/α

k−1/α





is C

and satisﬁes (25) on the region

|x| ≤ 2R, 0 ≤ t ≤ c

(ρ, k, α)u

1+nα





2kα

provided α ∈ (1/k, 1/(k −1)). If α ≥ 1/(k −1) then u is not C

but acts as

a barrier for smooth, strictly convex solutions of Eq. (6), hence also for the

viscosity solutions constructed in Theorem 15.

30 BEN ANDREWS

Proof. The case α ∈ (1/k, 1/(k − 1)) follows by direct computation. In

the case α ≥ 1/(k −1) the same computation gives the result except when

|x| = R − c

/R)

2kα

R/u

1+nα

t, where u is not C

. A smooth convex

hypersurface lying outside the graph of u and meeting the graph at such a

point must have K = 0, so the barrier condition is veriﬁed. 

To use these barriers in the comparison principle, we need to check that

the viscosity solution {M

} stays away from the boundary of the barrier

produced in Lemma 24. Fix x in the interior of M

and ε > 0 small, and

choose u

smaller than the smallest radius of curvature of N

, and place

the origin at the point x − (u

+ ε)ν

. Choose R suﬃciently small so that

the distance from x to the boundary of F

is at least 3R. Let

be the

hypersurface given by

n−k

i=1

= u







1 − c

(k, α)





i=n−k+1

− 1





k+1−1/α

k−1/α







for

i=n−k+1

≤ 4R

. Then

lies entirely within M

. Furthermore, for

each y ∈ ∂

, the sphere B

c(ρ,k,α) min{u

,R}

(y) is enclose d by M

for some

constant c. Deﬁne

to be the hypersurface

n−k

i=1

= u







1 − c





i=n−k+1

+ c





2kα

1+nα

− 1





k+1−1/α

k−1/α







for

i=n−k+1

≤ 4R

. For t < c(ρ, n, k, α) min{u

, R}

1+nα

we have

∂

⊂

[

y∈∂

r(t)

(y)

where r(t)

1+nα

= (c min{u

, R})

1+nα

− (1 + nα) sup ρt. The comparison

principle implies that each of these balls is enclosed by M

, and therefore

that ∂

does not me et M

. It follows by the comparison principle that

remains entirely enclosed by M

on this time interval, and therefore that

h(ν

, t) ≥ h(ν

, t) −ε for all ε > 0, so that x has not moved during this time

interval.

Now we proceed to the second part of the Theorem, the case of α ≤ 1/k.

This also proceeds using barrier constructions. Let x ∈ ∂Ω

. Since σ

n−k

> 0,

there exists a hyperplane P supporting Ω

at x and an n − k-dimensional

subspace Γ of P such that Ω

is contained in the region

hx − y, P

⊥

i ≥

n−k

|π

(x − y)|

MOTION OF HYPERSURFACES BY GAUSS CURVATURE 31

Ω

is also clearly contained in the ball B

(Ω

)

(x). We now construct bar-

riers:

Lemma 25. Fix k, ρ, α ∈ (0, 1/k) and constants R > 0 and u

> 0. If

every y ∈ Ω

satisﬁes

(26) |y| ≤ R,

n−k

i=1

≤ u

then for 0 < t ≤ c(ρ, k, n, α) min{u

, R}

1+nα

we have for every y = (Y, η) ∈

Ω

⊂ R

n+1−k

× R

' R

n+1

|Y | ≤ u

− ct

1−kα

α−1

1−kα

α(n−k)

1−kα



(4R − |η|)

−

α(k+1)

1−kα

+ (4R + |η|)

−

α(k+1)

1−kα



where c depends only on ρ, k, n, and α. If α = 1/k and every y ∈ Ω

satisﬁes

(26), then for 0 < t ≤ c(ρ, k, n) min{u

, R}

1+n/k

and y = (Y, η) ∈ Ω

we have

|Y | ≤ u

− c

R exp



−c

n−k

−k



cosh



2k−1

n−k

|η|t

−k



where c

and c

depend only on ρ, k, and n.

Proof. A direct calculation shows that these regions are given by cylindrical

graphs satisfying the inequality

(27) ˙u ≥ −inf ρ(−u

)

(−u

)

α(k−1)

−α(n−k)

|x|

−α(k−1)



1 + (u

)



1−α(n+2)

which therefore act as outer barriers for solutions of Eq. (6). 

It follows that the supp ort function at every point must change: If we

ﬁx z ∈ S

, and place a cylindrical barrier outside the hypersurface M

and

passing through x, then we have

h(z, t) ≤ h(z, 0) − ct

1−kα

−

1+kα

1−kα

−

α(n−k)

1−kα

for α < 1/k, and

h(z, t) ≤ h(z, 0) − c

R exp



−c

n−k

−k



when α = 1/k. Theorem 7 then gives lower bounds on the speed ρS

−α

uniformly in z for each t > 0, by the same argument as given in the proof of

Theorem 8. Theorem 10 implies bounds b e low on all the principal curvatures

at each positive time, and Corollary 12 gives upper bounds on the principal

curvatures. Theorem 5.5 of [K] implies that the solution hypersurface is

smooth and strictly convex for suﬃciently small positive time s, and the

result of Theorem 23 follows from Theorem 13. 

32 BEN ANDREWS

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34 BEN ANDREWS

[U2] , Complete noncompact self-similar solutions of Gauss curvature ﬂows I.

Positive powers, Math. Ann., 311 (1998), 251-274.

Received Octobe r 28, 1998. This research was partially supported by an Australian Re-

search Council Q EII Fellowship.

Australian National University

A.C.T. 0200

Australia

E-mail address: andrews@maths.anu.edu.au

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

SUMMATION OF FORMAL SOLUTIONS OF A CLASS OF

LINEAR DIFFERENCE EQUATIONS

B.L.J. Braaksma, B.F. Faber, and G.K. Immink

Dedicated to the memory of W.A. Harris, Jr

We consider diﬀerence equations y(s+1) = A(s)y(s), where

A(s) is an n × n-matrix meromorphic in a neighborhood of

∞ with det A(s) 6≡ 0. In general, the formal fundamental so-

lutions of this equation involve gamma-functions which give

rise to the critical variable s log s and a level 1

. We show

that, under a mild condition, formal fundamental matrices of

the equation can b e summed uniquely to analytic fundamental

matrices represented asymptotically by the formal fundamen-

tal solution in appropriate domai ns.

The method of proof is analogous to a method used to

prove multi-summability of formal solutions of ODE’s. Start-

ing from analytic lifts of the formal fundamental matrix in

half planes, we construct a sequence of increasingly precise

quasi-functions, each of which is determined uniquely by its

predecessor.

1. Introduction.

This paper is concerned with summability of formal solutions of linear ho-

mogeneous diﬀerence equations. We consider the system

y(s + 1) = A(s)y(s),(1.1)

where s is a complex variable, y(s) ∈ C

, and A(s) an n × n-matrix, mero-

morphic at inﬁnity, det A(s) 6≡ 0. For some p ∈ N Equation (1.1) has a

formal fundamental matrix solution of the form

Y(s) =

H(s)s

Λs

G(s)

,(1.2)

with

H(s) ∈ End(n, C[[s

−1/p

]]), det

H(s) 6≡ 0, Λ =

j=1

where λ

∈

and I

is the n

× n

-identity matrix, G(s) =

j=1

(s)I

where g

(s) ≡ 0

or g

(s) is a polynomial in s

1/p

of degree at most p with g

(0) = 0, and

L =

j=1

, L

= c

+ N

with c

∈ C and N

an n

× n

-nilpotent

matrix, and with n

+ n

+ · · · + n

= n.

36 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

The purpose of this pap er is to sum the entries of

H(s) on certain un-

bounded domains D in order to obtain uniquely characterizable analytic

fundamental matrix solutions

Y(s) = H(s)s

Λs

G(s)

, with

H(s) ∼

H(s), s → ∞ on D.

(1.3)

Any solution of (1.1) on D can be written as Y(s)P (s) where P (s) is a

1-periodic C

-valued function.

If the factor s

Λs

does not appear in the formal fundamental matrix, i.e.,

if all λ

’s vanish, the formal fundamental matrix resembles that of a homo-

geneous linear diﬀerential system. Formal power series solutions of mero-

morphic diﬀerential equations can be summed by means of a method known

as multisummation. With such an equation one can associate so-called ‘lev-

els’, positive rational numbers k

, . . . ,k

, and corresponding ‘critical vari-

ables’ s

, . . . ,s

, which play a crucial part in the summation process.

Multi-summation is a particular case of accelero-summation (see [Eca87]),

involving only elementary accelerations. There exist various equivalent deﬁ-

nitions of multisummability (see Deﬁnition 2). It can be formulated in terms

of Borel and Laplace transforms (cf. [MR91]), or in a more abstract way

(cf. [MR92]). In [Bal94] Balser presented yet another deﬁnition. Multi-

summability of solutions of both linear and nonlinear meromorphic diﬀeren-

tial equations has been proved both by using Borel-Laplace techniques (se e

[Bra91] and [Bra92]) and in a way based on the deﬁnition of Malgrange

and Ramis (see [BBRS91], [RS94], [Bal94], [Tov96], and [BIS]).

Two of the most important features that distinguish linear diﬀerence

equations from linear diﬀerential equations are:

(i) The solution space of a homogeneous linear diﬀerence equation is lin-

ear over the 1-periodic functions instead of C-linear as in the case of

homogeneous linear diﬀerential equations.

(ii) The occurrence of the factor s

Λs

, that does not appear in formal solu-

tions of diﬀerential equations.

If the factor s

Λs

does not appear in the formal fundamental matrix, or,

more generally, if all λ

’s are equal, then all entries of

H(s) are multi-

summable in all but at most a countably inﬁnite numbe r of directions. This

was shown in [BF96] by means of Borel-Laplace techniques in the spirit of

the work of Ecalle [Eca85]. With the same techniques multisummability

of formal solutions of a class of non-linear diﬀerence equations was proved

there.

If not all λ

’s are equal, some of the entries of

H(s) may not be multi-

summable in any direction. This is due to the fact that, in this case, one of

the critical variables is s log s, which is not a rational power of s.

Following Ecalle (cf. [Eca85]), one might set out to sum the formal so-

lutions by accelero-summation, using Borel and Laplace transforms. For a

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 37

particular class of linear diﬀerence equations, accelero-summability of the

formal solutions was established in [Imm]. Ecalle’s method involves the

study of a convolution equation, obtained from the equation satisﬁed by

by means of a formal Borel transformation in the variable s log s, which does

not look very inviting. In the present paper we take a diﬀerent approach,

similar to the method employed in [BIS] to sum formal solutions of lin-

ear diﬀerential equations (cf. Theorem 13). Our starting point is the ‘main

asymptotic existence theorem’ for diﬀerence equations (Theorem 6), which

says that

H(s) can be lifted on half planes in C

∞

, bounded by the real or

imaginary axis, to an analytic matrix H(s) such that (1.3) deﬁnes an analytic

fundamental matrix Y(s) of the diﬀerence equation. With the equation we

associate certain levels 0 < k

< · · · < k

= 1, that can be extracted from

the formal fundamental solution, as well as a level 1

if not all λ

’s are equal

(see Deﬁnitions 3 and 5). We choose a covering of a neighbourhood of ∞ in

∞

by appropriate half planes and, on each half plane a fundamental sys-

tem of (1.1) represented asymptotically by the formal fundamental system

(1.2). In several steps, modifying the solutions by exponentially small func-

tions of increasing order at each subsequent step, we construct a sequence

of so-called k

-precise quasi-functions, j = 1, . . . , r. If the e quation does not

possess a level 1

, this procedure yields the multi-sum, or (k

, . . . , k

)-sum

of the formal solution (Theorem 13).

If the equation does possess level 1

, the ﬁnal step is more delicate than

the preceding ones. This is due to the relative ‘closeness’ of the levels 1 and

and the transcendental nature of the critical variable s log s. In order to

end up with a unique sum, we need to consider domains that are strictly

smaller than half planes, but suﬃciently large to exclude the existence of ﬂat

solutions of the diﬀerence equation satisﬁed by

H, with a dominant factor

of the form s

(λ

−λ

, with λ

6= λ

. Here we shall consider domains of the

type {s ∈ C

∞

| arg s ∈ ((h − 1)π, (h + 1)π), (−1)

<{s(log s + iθ)} > 1},

with θ ∈ R, h ∈ Z (cf. Figures 2-5). On the union of two such domains with

the same h we can deﬁne a sum H(s) of

H(s) if a certain generic condition

is satisﬁed (cf. Section 7). By means of (1.3) we obtain a unique analytic

fundamental matrix of the diﬀerence equation (Theorem 18).

In order to illustrate the particular properties of diﬀerence equations with

level 1

, we e nd this introduction with a simple example.

Example 1. Consider the equation

h(s + 1) − as

−1

h(s) = s

−1

with a ∈ R, a > 0(1.4)

which can be transformed into the matrix equation





(s + 1) =



a/s 1/s

0 1





(s).

38 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

(1.4) belongs to a class of equations that was discussed in [BH75] and in

great detail by Ecalle in [Eca85, §3.6] and later by Immink in [Imm]. It

has a unique formal solution

h =

n≥1

−n

. Let ˆu(t) :=

n≥1

(n−1)!

n−1

the formal Borel transform of

h. The power series ˆu formally satisﬁes the

convolution equation

−t

u(t) − a(1 ∗ u)(t) = 1.

This equation has the unique analytic solution

u(t) = e

t−a

Thus ˆu(t) coincides with the Taylor series at t = 0 of this function and is

actually a convergent power series which extends to a holomorphic function

on C. The convergence of ˆu implies that

h is 1-Gevrey. By using Cauchy’s

formula for the coeﬃcients in a convergent Taylor series, one may derive the

more precise estimate

| ≤ K



log n



, ∀n ≥ 2, for some K, A > 0.

This type of estimate is typical of diﬀerence equations possessing a level 1

(cf. [Imm88]).

The function u(t) = e

t−a

is bounded in the left half plane and, conse-

quently,

h(s) is 1-summable in all directions in (

3π

). The 1-sum h

(s) is

analytic on the sector 0 < arg s < 2π and is a solution of (1.4).

In the right half plane u(t) grows faster than exponentially of any order

on the horizontal strips {t ∈ C | <t > 0, =t ∈ (−π/2, π/2) mod 2π}, but on

the strips

{t ∈ C | <t > 0, =t ∈ (π/2, 3π/2) mod 2π}

it decreases faster than exponentially of any order. Hence, the functions

r,n

(s) :=

−st

u(t)dt, n ∈ Z,(1.5)

with C

a path from 0 to +∞+iθ, θ ∈ (π/2+2nπ, 3π/2+2nπ) (see Figure 1)

are well deﬁned and satisfy (1.4).

π/2 + 2nπ

3π/2 + 2nπ

Figure 1. Contour C

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 39

The functions h

r,n

all have the asymptotic expansion

h as s → ∞, −

arg s <

, uniformly on closed subsectors (cf. also [BH75]). However, it

can be shown that the h

r,n

are not 1-sums. See also [vdPS97, Chapter 11].

In order to characterize these solutions by means of their asymptotic

behaviour, we have to consider this asymptotic behaviour on regions other

than sectors, namely regions of the form

D(θ) := {s ∈ C | <{s(log s + iθ)} > 1}, θ ∈ R,

see Figures 2-5.

-10

-5 5

-10

-5

Figure 2. Region D(0).

-10

-5 5

-10

-5

Figure 3. Region D(

-10

-5 5

-10

-5

Figure 4. Region D(π).

-10

-7.5 -5

-2.5 2.5

5 7.5

-10

-7.5

-5

-2.5

2.5

7.5

Figure 5. The regions ‘rotate’

clockwise with increasing θ.

40 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

Proposition. For any θ ∈ (2nπ, 2(n + 1)π), there exist K, A > 0 such that



r,n

(s) −

N−1

n=1

−n



≤ KA

(N!)|s|

−N

, ∀s ∈ D(θ), ∀N ∈ N.(1.6)

This proposition has been proved by Borel-Laplace methods in [Fab97].

According to a theorem by Immink in [Imm96], h

r,n

is uniquely determined

by the ab ove property.

2. Preliminaries.

By Argz we denote the principal argument of z ∈ C\{0}; we take Argz ∈

(−π, π]. The Riemann surface of the logarithm will be denoted by C

∞

For α, β ∈ R we denote by S(α, β) the open sector {s ∈ C

∞

| α < arg s <

β}, and by S[α, β] the closed sector {s ∈ C

∞

| α ≤ arg s ≤ β}. Similarly,

S[α, β) and S(α, β] denote half-open sectors. For µ ∈ Z we deﬁne

:= S((µ − 1)π/2, (µ + 1)π/2); H

:= S[(µ − 1)π/2, (µ + 1)π/2].

Throughout this paper, by an upper half plane, a fourth quadrant, etcetera,

we understand a lift of the upper half plane, the fourth quadrant, etcetera,

from the complex plane to the Riemann surface of the logarithm. A sector

will always be a sector of C

∞

with vertex at the origin.

By deﬁnition, a neighbourhood of ∞ in a sector S (S not necessarily

open) is an op e n subset U of S, such that, for any closed subsector S

of S

with aperture ≤ π, we can ﬁnd s

∈ S such that s

+ S

⊂ U. In particular,

iµπ/2

+ H

, µ ∈ Z, R > 0, is a neighbourhood of ∞ both in H

and in

If we write f(s) = O(g(s)) or f(s) = o(g(s)) as s → ∞ on a sector S, we

mean that f and g are functions deﬁned on a neighbourhood U of ∞ in S,

and that the O or o relation holds uniformly, as s → ∞, on the intersection

of U and any closed subsector of S.

Similarly, if

f(s) =

j≥0

−j/p

where p > 0, and if S is a s ector, then

f(s) ∼

f(s), s → ∞ on S, means the following: f is an analytic function on

a neighbourhood U of ∞ in S and for any closed subsector S

⊂ S and any

N ∈ N, we can ﬁnd positive constants R and C

,R,N

, such that



f(s) −

N−1

j=0

−j/p



≤ C

,R,N

|s|

−N/p

, ∀s ∈ S

∩ U, |s| > R.(2.1)

The set of such functions f with an asymptotic expansion

f on S as above

will be denoted by A(S).

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 41

In accordance with the above, when we write f(s) ∼ 0, s → ∞ on S, we

mean that f(s) = o(s

−N

), s → ∞ on S, for any N ∈ N.

Suppose ˆu(s) =

j=0

(s)(log s)

with

∈ C[[s

−1/p

]] for j = 0, . . . , m.

If we write u(s) ∼ ˆu(s), s → ∞ on S, we mean that there exist analytic

functions h

, j = 0, . . . , m, on a neighborhood of ∞ in S such that u(s) =

j=0

(s)(log s)

, with h

(s) ∼

(s), s → ∞ on S.

If f ∈ A(S) such that (2.1) holds and if there exist k > 0, and K

, A

> 0 such that, for each N,

,R,N

≤ K





then we call f a k-Gevrey function on S with respect to the family

and we write f ∈ A

(1/k)

(S). Note that

is an example of a ‘convenient

family’, ac cording to the terminology introduced by Malgrange in [Mal95].

Any f ∈ A

(1/k)

(S) has an asymptotic expansion

f(s) =

∞

j=0

−j/p

with the a

satisfying

| ≤ KA





, ∀j > 0,

for some positive K and A. Such a formal series

f will be called a Gevrey

series (in s

−1

) of order 1/k with respect to the family

, and

C[[s

−1/p

]]

denotes the set of such series.

In the sequel all Gevrey functions and Gevrey series will be with respect

to the family

with p as in (1.2), and we omit the references to this

family in our notations.

A function f deﬁned on a neighbourhood of ∞ in a sector S is exponen-

tially small of order k > 0 on S if for any closed subsector S

of S there

exists a p os itive constant c such that f(s) = O(e

−c|s|

), s → ∞ on S

. If

this holds for all positive c then f is said to be supra-exponentially small of

order k on S. The set of all analytic functions on a neighbourhood of ∞ in

S which are exponentially or supra-exponentially small of order k on S will

be denoted by A

≤−k

(S) and A

<−k

(S) respectively. If S = S(α, β) then we

will also write these latter sets as A

≤−k

(α, β) and A

<−k

(α, β). Similarly if

S = S(α, β] etc. If f and g both are in A

(1/k)

(S), and f and g have the

same asymptotic expansion, then it can be shown that their diﬀerence f − g

is in A

≤−k

(S) (cf. [Mal95]).

Let l > 0 and S be an open sector. Let {S

}

i∈I

be a covering of S

consisting of open sectors and let f

(i)

∈ A(S

), i ∈ I, such that f

)

−f

)

∈

≤−l

∩ S

) for any i

, i

∈ I with S

∩ S

6= ∅. These data determine

an l-precise quasi-function on S. We identify two such sets of data

({f

(i)

}

i∈I

; {S

}

i∈I

) and ({g

(j)

}

j∈J

; {

}

j∈J

) if f

(i)

− g

(j)

∈ A

≤−l

∩

)

where i ∈ I, j ∈ J such that S

∩

6= ∅. They deﬁne the same l-precise

42 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

quasi-function on S and we write (A/A

≤−l

)(S) for the set of these l-precise

quasi-functions on S. Indeed, if we identify the interval I = (a, b) in R

and the sector S(a, b), then A and A

≤−k

, k > 0, can be considered as

sheaves on R and then A/A

≤−l

is the quotient sheaf. Also elements of

(A/A

≤−l

)

(S)[log s] with n ∈ N will be called l-precise quasi-functions on

All elements in a representative {f

(i)

}

i∈I

of f ∈ (A/A

≤−l

)

(S)[log s] have

the same asymptotic expansion

f and this expansion is independent of the

chosen representative. Therefore we may write f(s) ∼

f(s), s → ∞ on S,

without causing confusion.

Similarly if l > k > 0 we deﬁne (A

(1/k)

≤−l

)(S) as the set of l-precise

quasi-functions f which have representatives f

on S

as above with f

∈

(1/k)

) for all i ∈ I.

f ∈ C[[s

−1/p

]]

then there exists a unique f ∈ (A

(1/k)

≤−k

)(C

)

such that f ∼

f, where C

denotes the Riemann surface of s

1/p

. This f will

be denoted by T

−1

f (cf. [MR92, Cor. (1.8)]).

Let f ∈ (A/A

≤−l

)

(S)[log s]. The ‘restriction’ of f to an open subsector

of S, denoted by f|

, is deﬁned as follows: Suppose {f

(i)

}

i∈I

is a rep-

resentative of f with respe ct to a covering {S

}

i∈I

of S. Then f|

is the

element of (A/A

≤−l

)

)[log s] deﬁned by {f

(i)

∩S

}

i∈I

, where f

(i)

∩S

is the restriction of f

(i)

to a neighb o urhood of ∞ in S

∩ S

Deﬁnition 2. Let 0 < k

< · · · < k

, and let

f ∈ C[[s

−1/p

]]

1/(pk

)

. Fur-

thermore, let S

⊃ . . . ⊃ S

be a nested sequence of open sectors, where

has ape rture larger than π/k

, i = 1, . . . , r and S

has ape rture at most

2pπ. We say that

f is (k

, . . . , k

)-summable on (S

, . . . , S

) if there exist

∈ (A

(1/k

)

≤−k

i+1

)(S

), i = 1, . . . , r − 1, and f

∈ A

(1/k

)

), such that

i+1

≡ f

i+1

mod A

≤−k

i+1

, i = 0 , . . . , r − 1 where f

= T

−1

f. We call f

the (k

, . . . , k

)-sum of

f on (S

, . . . , S

), and we have f

(s) ∼

f(s), s → ∞

on S

According to the ‘relative Watson lemma’ ([MR92, Prop. (2.1)]) f

i+1

is completely determined by f

and S

i+1

, i = 0, . . . , r − 1. Hence the

, . . . , k

)-sum of

f on (S

, . . . , S

) is uniquely deﬁned. We may

extend the deﬁnition of multisummability in an obvious way to the case

that

f is an n-vector or an n × n-m atix with elements in C[[s

−1/p

]]

1/(pk

)

Deﬁnition 3. Let f(s) = s

2πisb+q(s)

, with d ∈

Z, b ∈ C, q(s) iden-

tically zero or a polynomial in s

1/p

without constant term and of degree at

most p−1, and γ ∈ C. We will say that f(s) is of level 1

if d 6= 0, of le vel

1 if d = 0, b 6= 0, of level k with k ∈ {

, . . . ,

p−1

} if d = b = 0, q(s) 6≡ 0 and

q(s) has degree pk in s

1/p

, and of level 0 if d = b = 0 and q(s) ≡ 0.

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 43

Let f be of level k ∈ (0, 1], so d = 0, 2πisb + q(s) 6≡ 0. A closed interval

[σ−π/k, σ] will be called a Stokes interval of level k of f if f ∈ A

≤−k

(σ−

π/k, σ). So if k = 1 then σ ≡ π − Argb mod 2π whereas if 0 < k < 1 and

q(s) = ωs

+ o(s

), s → ∞, ω 6= 0, then kσ ≡

π − Arg ω mod 2π.

If f is of level 1

we will, in Section 7, associate with it a certain Stokes

number. This number is connected with curves that separate regions of

growth from regions of decay. All these curves have the limiting directions

mod π.

If f is of level 1

, we have f(s) = exp(ds log s(1 + o(1))), s → +∞ with

d 6= 0, and so it grows or decays faster than exponentially of order 1 on R

but slower than any higher exponential order.

Deﬁnition 4. For any (i.e., not necessarily open) sector S we will write

f ∈ A

≤−1

(S) to express that f is analytic on a neighbourhood U of ∞ in

S, and that for any c losed subsector S

of S, there exists a positive constant

c (dep e nding on S

) such that f(s) = O(e

−c|s| log |s|

), uniformly as s → ∞ on

∩ U.

We deﬁne 1

-precise quasi-functions by replacing l by 1

in the deﬁnition

of l-precise quasi-functions above.

So, for e xample, e

ds log s

∈ A

≤−1

) if d <0. And if p(s)=

j≥0

2πisj

is an analytic 1-periodic function on {s ∈ C|=s > R} for some R > 0, then

p(s)e

ds log s

∈ A

≤−1

∩ H

) if d < 0.

With Equation (1.1) we associate levels, and with each level certain Stokes

intervals or numbers. For this purpose we rewrite the formal fundamental

matrix solution (1.2) as follows:

Y(s) =

U(s)F(s),(2.2)

where

U(s) =

H(s)s

, N =

j=1

and F(s) = s

Λs

G(s)

, C =

j=1

The columns ˆy

(s) (l = 1, . . . , n) of

Y(s) form a formal fundamental

system of solutions {ˆy

}

l=1

and we have

ˆy

(s) = f

(s)ˆu

(s), f

(s) = s

2πisb

(s)

,(2.3)

where ˆu

(s) ∈ C

[[s

−1/p

]][log s] is the l-th column of

U(s), and, furthermore,

if 0 < l − (n

+ . . . + n

j−1

) ≤ n

, then d

= λ

, b

∈ C and q

(s) ≡ 0 or q

(s)

is a polynomial in s

1/p

without constant term and of degree at most p − 1

such that 2πisb

+ q

(s) = g

(s) and γ

= c

. Without loss of generality, we

may assume that <b

∈ [0, 1), l = 1, . . . , n.

We use the following abbreviations (cf. (2.3)): f

:= f

−1

, d

− d

, b

:= b

− b

, q

:= q

− q

, γ

:= γ

− γ

. We write κ

for

the level of f

44 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

Deﬁnition 5. The levels of Equation (1.1) are the levels of the functions

2πisj

(s), j ∈ Z, m, l ∈ {1, . . . , n}. Let k ∈ {

, . . . ,

p−1

, 1}. The Stokes

intervals of level k of the equation are the Stokes intervals of level k of the

functions e

2πisj

(s), j ∈ Z, m, l ∈ {1, . . . , n}.

Taking j = 0 we see that all the κ

are levels of the equation. Moreover,

0 and 1 always are levels of the equation (take m = l and then j = 0 and

j 6= 0, respectively). By 0 < k

< · · · < k

= 1 we denote the increasing

sequence of levels of the equation in the interval (0, 1]. If =b

6= 0 for some

m and l then there are inﬁnitely many Stokes directions (endpoints of Stokes

intervals) −Arg(b

+ j) mod π, j ∈ Z which cluster at 0 mod π.

The following theorem is the counter part in the theory of linear diﬀer-

ence equations of the ‘main asymptotic existence’ theorem in the theory of

diﬀerential equations.

Theorem 6. Let l ∈ {1, . . . , n} and ˆy

(s) = f

(s)ˆu

(s) be a formal solution

of (1.1) of the form (2.3) with ˆu

(s) ∈ C

[[s

−1/p

]][log s].

Then for any µ ∈ Z there exists an analytic solution y

(s) = f

(s)u

(s) of

the equation such that u

(s) ∼ ˆu

(s), s → ∞ on H

A proof of this theorem (for the case that no logarithmic terms appear

in ˆu(s)) can be found in [vdPS97]. It is based on the so-called quadrant

theorem, already stated by Birkhoﬀ and Trjitzinsky in [BT33], but made

rigorous by Immink in [Imm91].

3. Two auxiliary lemmas.

The following lemma gives information on the relation between two funda-

mental systems of solutions of Equation (1.1), which have the same asymp-

totic behaviour at ∞ on some sector.

Lemma 7. Suppose we have two fundamental matrix solutions of Equation

(1.1), Y = UF and Y

= U

F, such that U(s) ∼

U(s) and U

(s) ∼

U(s) for

s → ∞ on an open sector S, with F(s) and

U(s) as in (2.2). Let u

and u

l,1

be the l-th column of U and U

, respectively.

Then there exist analytic 1-periodic functions p

, m = 1, . . . , n, on a

neighbourhood of ∞ in S, such that

− u

l,1

m=1

Moreover,

(s)f

(s) ∼ 0, s → ∞ on S, ∀m ∈ {1, . . . , n}.

− u

l,1

∈ (A

≤−k

)

(S) for some k > 0 (including k = 1

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 45

then

∈ A

≤−k

(S), ∀m ∈ {1, . . . , n}.

Proof. Let

U := U − U

. Then

UF = UFP, or, equivalently, FPF

−1

= U

−1

for some 1-periodic analytic matrix function P = (P

), on a neighbourhood

of ∞ in S. From the diagonal form of the matrix F it follows that if p

the element in the m-th row and l-th column of P, then

= (m-th row of U

−1

)(u

− u

l,1

As U(s) ∼

U(s) we have U

−1

(s) ∼

−1

(s) = s

−N

−1

(s). Hence any entry

of U

−1

is of order O(s

(log s)

), s → ∞ for some µ, ν ∈ Z. Since u

(s) −

l,1

(s) ∼ 0 as s → ∞ on S, we thus ﬁnd that

(s)f

(s) ∼ 0, s → ∞ on S, ∀m ∈ {1, . . . , n}.

Similarly, we see that p

∈ A

≤−k

(S), if u

− u

l,1

∈ (A

≤−k

)

(S) for some

k > 0, including k = 1

. 

The next lemma yields more information on the asymptotic behaviour of

the functions p

in the previous lemma.

Lemma 8. Let S := S(α

, α

) be an open sector with 0 < α

− α

≤ π.

Let p(s) be an analytic, 1-periodic function on a neighbourhood of ∞ in

S, and let f be a function of level k ∈ {0, 1/p, . . . , 1, 1

} as in Section 2:

f(s) = s

2πisb+q(s)

with <b ∈ [0, 1). Assume g(s) := p(s)f(s) ∼ 0,

s → ∞ on S. Let H be an upper or lower half plane in C

∞

which has a

nonempty intersection with S.

Then g ∈ A

≤−k

(S) if k > 0 and g ∈ A

≤−1

(H) if k = 0. If ν denot es

some integer we have:

1) If k = 0: If α

< νπ < α

then p = g = 0.

2) If 0 < k < 1:

Then there exists c ∈ C such that g − cf ∈ A

≤−1

(H) and cf ∈

≤−k

(S). If α

< νπ < α

then p(s) = c. If g ∈ A

<−k

(S) then

g ∈ A

≤−1

(H).

3) If k = 1:

If p 6= 0 then with H corresponds an integer N such that p(s) ∼

2πiNs

as |=s| → ∞ on H where p

6= 0. If b ∈ R

∗

then g ∈

≤−1

(H) and if moreover α

< νπ < α

then p = g = 0. If α

≤

νπ ≤ α

and (−1)

=b < 0 then p = g = 0.

Next suppose

(I) α

≤ νπ ≤ α

and (−1)

=b > 0.

(II) (α

, α

) ⊂ (β

, β

) where (β

, β

) does not contain a Stokes interval

of e

2πisj

f(s) of level 1 for any j ∈ Z.

Then there exist analytic 1-periodic functions p

and p

−

such that

p = p

+ p

−

and p

f ∈ A

≤−1

(α

, β

) and p

−

f ∈ A

≤−1

(β

, α

). If

46 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

= νπ then p

f ∈ A

≤−1

[νπ, β

) and similarly if α

= νπ then

−

f ∈ A

≤−1

(β

, νπ].

4) If k = 1

If α

< (ν +

)π < α

, then p = g = 0. If νπ ≤ α

< α

≤

(ν +

)π or (ν −

)π ≤ α

< α

≤ νπ then g ∈ A

≤−1

[νπ, α

) and

g ∈ A

≤−1

(α

, νπ] respectively. If f 6∈ A

≤−1

(S) then p = g = 0.

5) If k ≤ 1 and g ∈ A

<−1

(S) then g = 0.

Proof. We will give the proof for the cases (i) 2hπ < α

< α

< (2h + 1)π,

(ii) α

= 2hπ and (iii) α

< 2hπ < α

for some h ∈ Z. The other cases can

be treated similarly.

We may choose H = H

4h+1

. So p(s) and g(s) are analytic on H for

=s > R for some R > 0. Put z = e

2πis

and P (z) := p(s). Then |z| = e

−2π=s

and we have expansions

P (z) =

∞

j=−∞

if 0 < |z| < e

−2πR

, p(s) =

∞

j=−∞

2πisj

if =s > R.

(3.1)

In case (iii) p is an entire function. So then (3.1) holds with p

= 0 if j < 0

and R may be replaced by −∞. We now treat separately the diﬀerent cases

of the lemma.

Ad 1) We have f (s) = s

. Hence p(s) = s

−γ

g(s) ∼ 0 as s → ∞ on

S. The 1-periodicity of p(s) then implies that p(s) ∼ 0 as =s → ∞, so

P (z) → 0 as z → 0. Therefore p

= 0 if j ≤ 0 and p, g ∈ A

≤−1

(H). In case

(iii) also p(s) ∼ 0 as =s → −∞, so P (z) → 0 as z → ∞. Hence P = 0 and

so g = p = 0.

Ad 2) For any closed sector S

⊂ (S ∩ H) and any ρ > R, there exist

positive constants K and a such that

|p(s)| = |g(s)f (s)

−1

| ≤ K exp(a|s|

), if s ∈ S

and =s ≥ ρ.

From the 1-periodicity of p(s) and the fact that k ∈ (0, 1) it follows that

|p(s)| ≤ K exp(a

=s), ∀=s ≥ ρ, for some a

∈ (0, 2π), if we choose ρ suﬃ-

ciently large. This implies P (z) = o(z

−1

), z → 0. Hence p

= 0 if j < 0

in (3.1). With c := p

we get p(s) − c ∈ A

≤−1

(H) and g − cf ∈ A

≤−1

(H).

If f 6∈ A

≤−k

(S ∩ H) then c = 0 since otherwise g is unbounded in a neigh-

bourhood of ∞ in S. So in cases (i) and (ii) we have cf ∈ A

≤−k

(S). Also

c = 0 if g ∈ A

<−k

(S) and therefore g ∈ A

≤−1

(H). In case (iii) we have

moreover P (z) = o(z) as z → ∞ and so P (z) = p(s) ≡ c. If c 6= 0 then

g = cf ∈ A

≤−k

(S) as g ∼ 0 in S.

Ad 3) The fact that f is of level 1 implies that d = 0, b 6= 0.

We have p(s) = g(s)f(s)

−1

= O(1) exp(−2πis(b + o(1))) as s → ∞ on

S ∩ H and therefore P (z) = O(z

b+o(1)

) as z → 0. So if p 6= 0 then there

exists N ∈ Z such that p

= 0 for all j < N and p

6= 0 in (3.1). Then

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 47

p(s) = p

2πiNs

(1+o(1)) and g(s) = p

2πi(b+N+o(1))s

(1+o(1)) as =s → ∞.

As g(s) ∼ 0 as s → ∞ in S it follows that g ∈ A

≤−1

(S ∩H) where S ∩H = S

in cases (i) and (ii). Moreover, if =b = 0 then b+N > 0 and so g ∈ A

≤−1

(H).

In cases (ii) and (iii) we see that if p 6= 0 and =b < 0 then g(s) → ∞ on

arg s = 2hπ +  for  suﬃciently small p ositive. Hence p = g = 0 if =b < 0.

In case (iii) similar reasoning as above leads to p

= 0 for all j > M

with some M ∈ Z, M ≥ N, g(s) = e

2πi(b+M+o(1))s

+ o(1)) as =s → −∞

and g ∈ A

≤−1

(α

, 2hπ). In particular, if =b = 0 and p

6= 0 6= p

, then

the fact that g ∼ 0 in S(2hπ − , 2hπ + ) for some  > 0 implies that

N + b > 0 > M + b in contradiction with M ≥ N, and therefore p = g = 0.

So in case (iii) we have =b 6= 0 if p 6= 0 and so =b > 0. Consequently g is

exponentially small of order 1 in S(2hπ − , 2hπ + ) for some  > 0. Thus

g ∈ A

≤−1

(S).

Next consider the case that (I) and (II) are satisﬁed. Now ν = 2h and

only cases (ii) and (iii) with =b > 0 have to be considered. Let σ

(2h + 1)π − Arg(b + j) for all j ∈ Z. Then S(σ

− π, σ

) is a maximal sector

where e

2πisj

f(s) is exponentially small of order 1 and the behaviour of g on

S ∩ H implies that σ

− π ≤ 2hπ < α

≤ σ

. If σ

≥ β

then we see that

g ∈ A

≤−1

[2hπ, β

). Next suppose σ

< β

. As σ

increases monotonically

from 2hπ to (2h + 1)π as j increases from −∞ to +∞, there exists A ∈ Z

such that σ

< β

≤ σ

A+1

and A ≥ N . The condition on Stokes intervals

now implies that σ

− π ≤ β

Let p

−

(s) :=

j=N

2πisj

. Then p

−

(s) = O(e

2πisA

) on the lowe r

half plane and p

−

(s) = O(e

2πisN

) on the upper half plane. So p

−

f is

exponentially small of order 1 for arg s ∈ (σ

− π, 2hπ] and for arg s ∈

[2hπ, σ

). Hence p

−

f ∈ A

≤−1

(β

, α

). Furthermore, p

:= p − p

−

∞

j=A+1

2πisj

= O(e

2πis(A+1)

) on the upper half plane and as σ

A+1

≥ β

we see that p

f ∈ A

≤−1

[2hπ, β

). Moreover, p

f = g−p

−

f ∈ A

≤−1

(α

, α

)

and we conclude that p

f ∈ A

≤−1

(α

, β

Ad 4) Now f (s) = exp{ds(log s + O(1))}, s → ∞ with d 6= 0.

Since <(s log s) = <s log |s| − =s arg s, we have p(s) = g(s)f(s)

−1

O(exp{−d<s log |s| + O(s)}), as s → ∞ on S. As <s/=s is a nonzero

constant on any ray arg s = ψ 6∈

Z we see that if S contains such a ray on

which d<s > 0 then p(s) = O(1) exp(−N|=s|) for any N ∈ N and therefore

p = g = 0. In particular, if f 6∈ A

≤−1

(S) then there exists a ray where

d<s > 0, so p = g = 0.

It is now suﬃcient to consider the case that S belongs to a right half

plane and d < 0. Choose ε with 0 < ε < (α

− α

)/2 and ε < α

/2 − hπ.

Let ψ ∈ (2hπ, α

− 2ε). For any s with arg s = ψ there exists s

−

∈ H with

s − s

−

∈ N and arg s

−

∈ (α

− ε, α

− ε/2) if =s is suﬃciently large. Then

−

< =s cot(α

− ε). Thus

p(s) = p(s

−

) = O(1) exp[−d cot(α

− ε)=s log =s + O(s)].

48 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

From this and =s log =s = |s| sin ψ log(|s| sin ψ) = |s|(sin ψ log |s|+ O(1)) we

conclude that |p(s)| ≤ K

exp{−d cot( α

− ε) sin ψ|s| log |s| + K

|s|} if =s

is suﬃciently large where K

and K

are some positive constants. Using

g(s) = O(1)p(s) exp[−|ds|(cos ψ) log |s| + O(s)] and cos ψ − (sin ψ) cot(α

−

ε) = sin(α

− ε − ψ)/ sin(α

− ε) > sin ε/ sin(α

− ε) =: c

> 0 we see that

g(s) = O(1) exp(−c

|ds| log |s|)

if =s is suﬃciently large and arg s ∈ (2hπ, α

− 2ε). Thus we see that

g ∈ A

≤−1

[2hπ, α

). In case (iii) we get similarly g ∈ A

≤−1

(α

, 2hπ].

Furthermore, the 1-periodic function p is bounded on any bounded strip

parallel to the real axis intersected with S. Thus we obtain g ∈ A

≤−1

(S)

in case (iii).

Ad 5) If g(s) ∈ A

<−1

(S) and k ≤ 1 we deduce p(s) = g(s)/f(s) ∈

<−1

(S). T herefore p(s) = O(1)e

−c|s|

as s → ∞ for all c > 0. So P (z) =

O(z

) as z → 0 for all j. Hence P (z) ≡ 0 and so p = g = 0. 

4. A Gevrey property of solutions

Proposition 9. Let k

be the lowest positive level of (1.1). Then the ele-

ments of

H(s) are Gevrey series of order 1/k

. There exist fundamental

matrices Y

(µ)

(s) = H

(µ)

(s)s

Λs

G(s)

of (1.1) such that H

(µ)

(s) is a matrix

of k

-Gevrey functions on H

with H

(µ)

(s) ∼

H(s) on H

for all µ ∈ Z.

For any µ

∈ Z a representative of T

−1

H (cf. deﬁnition of T

−1

in Sec-

tion 2) on the covering {H

| µ = µ

, . . . , µ

+ 4p − 1} of C

is given by

(µ)

| µ = µ

, . . . , µ

+ 4p − 1}.

Let S be an open sector of aperture at most π and let ˆu

be given by (2.3)

for l = 1, . . . , n. Assume that f

is a solution of (1.1) such that v

∼ ˆu

S for l = 1, . . . , n. Then v

∈ (A

(1/k

)

(S)[log s]. Moreover, {f

}

l=1

is a

fundamental set of solutions of equation (1.1).

Proof. To prove the last statement, let V be the matrix with v

as l-th

column. Then V ∼

U on S, where

U as in (2.2). As det

U 6≡ 0, we also

have det V 6≡ 0. Thus Y := VF is a matrix solution of Equation (1.1) and

det Y 6≡ 0, i.e., it is a fundamental matrix solution.

According to Theorem 6 and the last statement of the proposition under

consideration we have fundamental matrices Y

(µ)

(s) = H

(µ)

(s)s

F(s), µ =

, . . . , µ

+ 4p − 1, with H

(µ)

(s) ∼

H(s), s → ∞ on H

H(s) as in (1.2).

Since e

2pπi

= H

µ+4p

we deﬁne H

(µ

+4p)

(s) = H

(µ

)

(se

−2pπi

), s ∈ H

+4p

Then H

(µ

+4p)

(s) ∼

H(se

−2pπi

) =

H(s), s → ∞ on H

+4p

. If s ∈ H

+4p

ζ := se

−2pπi

∈ H

, then s + 1 = (ζ + 1)e

2pπi

and

(µ

+4p)

(s) := H

(µ

+4p)

(s)s

F(s)

= H

(µ

)

(ζ)(ζe

2pπi

)

F(ζe

2pπi

) = Y

(µ

)

(ζ)P(ζ),

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 49

where P(ζ) = e

2pπi(ζΛ+L)

, with Λ and L as in (1.2), is a 1-periodic matrix

function, and det P (ζ) 6= 0. Hence Y

(µ

+4p)

(s) is a fundamental matrix.

Next we prove that the entries of H

(µ)

(s) are in A

(1/k

)

), µ = µ

, . . . ,

+4p. As the half planes H

, µ = µ

, . . . , µ

+4p−1, cover a neighbourhood

of ∞ on the Riemann surface of z

1/p

, it is, by [MR92, Theorem 1.6], suﬃ-

cient to prove that the entries of H

(µ+1)

(s) − H

(µ)

(s) are exponentially small

of order k

on H

∩H

µ+1

, µ = µ

, . . . , µ

+4p−1. If we denote by u

(µ)

(s) the

l-th column of H

(µ)

(s)s

, l = 1, . . . , n, µ = µ

, . . . , µ

+4p, this is equivalent

to proving that the diﬀerences u

(µ+1)

− u

(µ)

are in (A

≤−k

)

∩ H

µ+1

We have

(µ+1)

− u

(µ)

m=1

(µ)

for some 1-periodic functions p

on a neighbourhood of ∞ in H

∩ H

µ+1

Since u

(µ+1)

(s)−u

(µ)

(s) ∼ 0, as s → ∞ on H

∩H

µ+1

, we have p

(s)f

(s)∼

0, s → ∞ on H

∩H

µ+1

, for m = 1, . . . , n, according to Lemma 7. Lemma 8

now yields that p

∈ A

≤−k

∩ H

µ+1

), m = 1, . . . , n, so

(µ+1)

− u

(µ)

∈ (A

≤−k

)

∩ H

µ+1

Applying [MR92, Theorem 1.6], we conclude that

(µ)

∈ (A

(1/k

)

)[log s], l = 1, . . . , n, µ = µ

, . . . , µ

+ 4p − 1,

(µ)

is a k

-Gevrey function on H

and the elements of

H(s) are Gevrey se ries

of order 1/k

. Moreover, it follows that {H

(µ)

| µ = µ

, . . . , µ

+ 4p − 1} is

a representative of T

−1

Finally we prove the statement concerning the functions v

(s). It is suf-

ﬁcient to consider the case that S ⊂ (H

∪ H

). There exist 1-periodic

functions ˜p

analytic on a neighbourhood of ∞ in S ∩ H

, such that

− u

(µ

)

m=1

˜p

(µ

)

Since v

(s) − u

(µ

)

(s) ∼ 0, as s → ∞ on S ∩ H

, Lemma 7 and Lemma 8

now tell us that ˜p

∈ A

≤−k

(S ∩ H

), hence

− u

(µ

)

∈ (A

≤−k

)

(S ∩ H

The same holds with µ

replaced by µ

+ 1. It follows that v

∈

(1/k

)

(S)[log s], what had to be proven. 

50 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

5. Reﬁnement of chains of solutions.

Consider the fundamental matrix Y

(µ)

(s) of Proposition 9 for µ ∈ Z. Let

its c olumns be denoted by {f

(µ)

l,0

} for l = 1, . . . , n as in (2.3). Then u

(µ)

l,0

∈

(1/k

)

)[log s], u

(µ)

l,0

∼ ˆu

on H

and {u

(µ)

l,0

| µ = µ

, . . . , µ

+ 4p −

1} represents the k

-precise quasi-function corresponding to T

−1

ˆu

on the

covering {H

| µ = µ

, . . . , µ

+ 4p − 1} of C

for any µ

∈ Z.

In this section we show how these k

-precise quasi-functions can be reﬁned

to k

-precise quasi-functions with representatives {u

(µ)

l,1

} on an open sector

S(α

, β

) with aperture > π/k

such that (α

, β

) does not contain a Stokes

interval of level k

of the equation and such that {f

(µ)

l,1

}

l=1

again is a

fundamental system of (1.1) with the same asymptotic expansion as before.

This will be done by expressing the diﬀerences f

(µ)

l,0

− u

(µ+1)

l,0

) in terms of

suitable fundamental systems and distributing the terms that are f

times

an exponentially small factor of order k

, over u

(µ)

l,0

and u

(µ+1)

l,0

. The same

method can be applied to proceed from k

-precise quasi-functions u

(µ)

l,j−1

to k

j+1

-precise quasi-functions u

(µ)

l,j

corresponding to solutions f

(µ)

l,j−1

and

(µ)

l,j

of (1.1).

Proposition 10. Let 0 < k

< · · · < k

= 1 be the levels in (0, 1] of (1.1).

Let j ∈ {1, . . . , r}, and deﬁne k := k

. If j < r, then k

:= k

j+1

, otherwise

:= 1

. Let (α, β) be an open interval of length > π/k not containing a

Stokes interval of level k of (1.1). Let M and N be the integers, such that

(M − 1)

≤ α < M

< N

< β ≤ (N + 1)

. Deﬁne Γ

:= H

∩ S(α, β)

for µ = M, . . . , N.

Suppose that we have fundamental systems of solutions {f

(µ)

}

l=1

on Γ

for µ = M, . . . , N which satisfy for l = 1, . . . , n:

(i) u

(µ)

∈ (A

(1/k

)

(Γ

)[log s], u

(µ)

(s) ∼ ˆu

(s), s → ∞ on Γ

(ii) u

(µ+1)

− u

(µ)

∈ (A

≤−k

)

∩ H

µ+1

Then there exist fundamental systems of solutions {f

˜u

(µ)

}

l=1

for µ =

M, . . . , N such that for l = 1, . . . , n:

˜u

(µ)

− u

(µ)

∈ (A

≤−k

)

(Γ

), if µ ∈ {M, . . . , N }(5.1)

and

˜u

(µ+1)

− ˜u

(µ)

∈ (A

≤−k

)

∩ H

µ+1

), if µ ∈ {M, . . . , N − 1}.(5.2)

Moreover, for each l ∈ {1, . . . , n} the family of functions {˜u

(µ)

}

µ=M

deﬁnes

an element ˜u

in (A

(1/k

)

≤−k

)

(α, β)[log s], which is uniquely determined

by the properties of the ˜u

(µ)

mentioned above.

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 51

We prove the proposition subsequently for the cases k ∈ (0, 1) and k = 1.

Proof for k ∈ (0, 1).

We introduce the following sets:

−

(µ, l) = {m | κ

= k, f

∈ A

≤−k

(α, (µ+1)π/2)} if µ ∈ {M −1, . . . , N −

1};

(µ, l) = {m | κ

= k, f

∈ A

≤−k

(µπ/2, β)} if µ ∈ {M, . . . , N}. Obvi-

ously

−

(µ + 1, l) ⊂ St

−

(µ, l), and St

(µ − 1, l) ⊂ St

(µ, l).(5.3)

Because of the assumption that β − α > π/k the two sets St

−

(µ, l) and

(µ, l) are disjoint. Since f

= f

the following transitivity relation

holds:

j ∈ St

−

(µ, m) ∧ m ∈ St

−

(µ, l) ⇒ j ∈ St

−

(µ, l).(5.4)

Finally, let St(µ, l) := St

−

(µ, l) ∪ St

(µ, l), µ = M, . . . , N − 1. If µ ∈

{M, . . . , N − 1} then

= k and f

∈ A

≤−k

∩ H

µ+1

) ⇐⇒ m ∈ St(µ, l).(5.5)

We only give the proof that the left statement implies the right one since the

converse is trivial. The left-hand side implies that f

has a Stokes interval

[σ − π/k, σ] c ontaining (µπ/2, (µ + 1)π/2). Because of the assumptions of

the proposition we have either α < σ−π/k < β ≤ σ or σ −π/k ≤ α < σ < β

and therefore m ∈ St(µ, l).

For k < 1 the ﬁrst statement of Proposition 10 is an easy consequence of

the following two lemmas.

Lemma 11. Under the assumptions of Proposition 10 with k < 1 there exist

fundamental systems {f

(µ)

l,1

}

l=1

, µ = M, . . . , N, satisfying:

(i) u

(µ)

l,1

− u

(µ)

∈ (A

≤−k

)

(Γ

);

(ii) u

(µ+1)

l,1

− u

(µ)

l,1

m∈St

−

(µ,l)

(µ)

m,1

+ ψ

(µ)

l,1

, where c

(µ)

∈ C and

(µ)

l,1

∈ (A

≤−k

)

∩ H

µ+1

Proof. The proof goes by induction on µ. Deﬁne u

(M)

m,1

:= u

(M)

, m =

1, . . . , n. Next assume u

(M)

m,1

, . . . , u

(µ)

m,1

have been deﬁned for all m = 1, . . . , n

and some µ ∈ {M, . . . , N − 1}. In the remaining part of this section m will

always be understood to be in {1, . . . , n}. Fix l ∈ {1, . . . , n}.

From condition (i) of the prop osition it follows that u

(µ)

m,1

∼ ˆu

on Γ

Thus the functions f

(µ)

m,1

, m ∈ St

−

(µ, l), together with the functions

(µ+1)

, m 6∈ St

−

(µ, l), form a fundamental s ystem of solutions according

52 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

to Proposition 9. Hence, there exist 1-pe riodic analytic functions p

(µ)

, p

(µ+1)

1 ≤ m ≤ n, such that

(5.6) u

(µ+1)

− u

(µ)

l,1

m∈St

−

(µ,l)

(µ)

m,1

m∈St

(µ,l)

(µ+1)

m6∈St(µ,l)

(µ+1)

We have u

(µ+1)

− u

(µ)

l,1

= u

(µ+1)

− u

(µ)

+ u

(µ)

− u

(µ)

l,1

∈ (A

≤−k

)

∩ H

µ+1

and thus (by Lemma 7) we may conclude that each term in the sums on

the right-hand side of (5.6) belongs to this set. Next we apply Lemma 8 to

these terms, and ﬁnd:

• If m ∈ St

−

(µ, l) (resp. m ∈ St

(µ, l)): Then f

is of level k < 1,

and it is an element of A

≤−k

∩ H

µ+1

). So there exist complex

constants c

(µ)

(resp. d

(µ+1)

) such that (p

(µ)

(s) − c

(µ)

(s) (resp.

(µ+1)

(s) − d

(µ+1)

(s)) belong to A

≤−1

(H), where H is the upper

or lower half plane containing H

∩ H

µ+1

• If m 6∈ St(µ, l): Then κ

< k or κ

= k with f

6∈ A

≤−k

∩H

µ+1

)

or κ

> k with in all cases p

(µ+1)

∈ A

≤−k

∩ H

µ+1

). In the

last case Lemma 8 implies p

(µ+1)

∈ A

≤−k

∩ H

µ+1

). This also

follows in the ﬁrst two cases from Lemma 8-2 with c = 0.

So we have

(µ+1)

− u

(µ)

l,1

m∈St

−

(µ,l)

(µ)

m,1

m∈St

(µ,l)

(µ+1)

+ ψ

(µ)

l,1

with ψ

(µ)

l,1

a function in (A

≤−k

)

∩ H

µ+1

). Obviously, if we deﬁne

(µ+1)

l,1

:= u

(µ+1)

−

m∈St

(µ,l)

(µ+1)

then u

(µ+1)

l,1

satisﬁes the requirements.

So, if we have constructed u

(λ)

m,1

for m ∈ {1, . . . , n} and λ ∈ {M, M +

1, . . . , µ} (µ ≤ N − 1), then we can construct u

(µ+1)

l,1

for each l ∈ {1, . . . , n}

and the lemma follows by induction on µ. 

We next reﬁne the solutions of the previous lemm a to solutions which

satisfy (5.1) and (5.2) in Proposition 10.

Lemma 12. Let k < 1. Suppose the assumptions of Proposition 10 hold,

and furthermore, assume that u

(µ+1)

−u

(µ)

m∈St

−

(µ,l)

(µ)

+ψ

(µ)

for some constants c

(µ)

and a function ψ

(µ)

∈ (A

≤−k

)

∩ H

µ+1

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 53

Then there exist fundamental systems {f

˜u

(µ)

}

l=1

, µ = M, . . . , N , such

that

(i) ˜u

(µ)

− u

(µ)

m∈St

−

(µ,l)

˜c

(µ)

for some constants ˜c

(µ)

;

(ii) ˜u

(µ+1)

− ˜u

(µ)

∈ (A

≤−k

)

∩ H

µ+1

Proof. This lemma can also be proven by induction, but this time we start

from the other end of the covering {Γ

}

µ=M

of S(α, β): deﬁne ˜u

(N)

:= u

(N)

for all m ∈ {1, . . . , n}. Next suppose ˜u

(N)

, . . . , ˜u

(µ+1)

have been deﬁned

and possess the properties of the lemma for all m ∈ {1, . . . , n} and some

µ ∈ {M, . . . , N − 1}. Let l ∈ {1, . . . , n}. We have

˜u

(µ+1)

− u

(µ)

= ˜u

(µ+1)

− u

(µ+1)

+ u

(µ+1)

− u

(µ)

m∈St

−

(µ+1,l)

˜c

(µ+1)

m∈St

−

(µ,l)

(µ)

+ ψ

(µ)

Furthermore,

(µ+1)

− u

(µ)

j∈St

−

(µ,m)

(µ)

+ ψ

(µ)

From these two relations and properties (5.3) and (5.4) we obtain

˜u

(µ+1)

− u

(µ)

m∈St

−

(µ,l)

˜c

(µ)

where the ˜c

(µ)

are constants in C and

(µ)

∈ (A

≤−k

)

∩ H

µ+1

If we deﬁne

˜u

(µ)

:= u

(µ)

m∈St

−

(µ,l)

˜c

(µ)

then ˜u

(µ)

satisﬁes the requirements of the lemma. Again the lemma follows

by induction on µ. 

The ﬁrst statement of Proposition 10 follows from the previous lemmas

and the last statement is a direct consequence of the relative Watson lemma

referred to after De ﬁnition 2. We could also prove the uniqueness directly,

without reference to this lemma, along the same lines as in [BIS]. 

Proof for k = 1.

Recall that the Stokes intervals of level 1 of Equation (1.1) with for-

mal solution (2.3) are, by deﬁnition, the Stokes intervals of the functions

2πisj

(s), j ∈ Z, l, m ∈ {1, . . . , n}, that are of level 1. Taking l = m and

54 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

j 6= 0, we ﬁnd that [(h−1)π, hπ] is a Stokes interval of level 1 of the equation

for any h ∈ Z. Hence, due to the assumption that ( α, β) does not contain a

Stokes interval of level 1, the sector S(α, β) does not contain a lift of both the

positive and the negative real axis. From this and α < Mπ/2 < N π/2 < β

it follows that N − M ≤ 2 and at least one of the integers M and N has to

be odd.

We will prove the proposition for the cases M = −1 and N = 0 or N = 1,

that is α ∈ [−π, −π/2), β ∈ (0, π]. The other cases can be proven similarly.

Let l ∈ {1, . . . , n}. There exist 1-periodic analytic functions p

(−1)

(s),

m = 1, . . . , n, such that

(0)

− u

(−1)

m=1

(−1)

.(5.7)

By assumption (ii) of Proposition 10, we have u

(0)

−u

(−1)

∈ (A

≤−1

)

(−π/2, 0).

So, by Lemma 7, p

(−1)

∈ A

≤−1

(−π/2, 0), m ∈{1, . . . , n}. From Lemma 8

we conclude that

(−1)

∈ A

≤−1

(−π, 0) if κ

< 1,(5.8)

and

(−1)

∈ A

≤−1

(−π/2, 0) if κ

= 1

.(5.9)

If κ

= 1 and p

(−1)

6= 0 then according to Lemma 8-3 we have =b

≥ 0.

Moreover, this lemma tells us that

(−1)

∈ A

≤−1

(−π, 0) if κ

= 1, =b

= 0.(5.10)

Let t

, . . . , t

denote the numbers =b

, h ∈ {1, . . . , n} in decreasing order

of magnitude. We will use induction on τ ∈ {1, . . . , ν}.

If =b

= t

, then =b

≤ 0 for all m ∈ {1, . . . , n}. So if κ

= 1 and

(−1)

6= 0 then we know already that =b

≥ 0 and so =b

= 0 and (5.10)

applies. Therefore if =b

= t

we deﬁne

˜u

(−1)

= u

(−1)

≤1

(−1)

, ˜u

(0)

= u

(0)

Then from (5.7), (5.8), (5.9) and (5.10) it follows that f

˜u

(µ)

are solutions

of (1.1) with

˜u

(µ)

− u

(µ)

∈ (A

≤−1

)

(Γ

) if µ ∈ {−1, 0};

˜u

(0)

− ˜u

(−1)

∈ (A

≤−1

)

(−π/2, 0).

(5.11)

Next let τ ∈ {2, . . . , ν}, =b

= t

and suppose that for all m ∈ I(l) :=

{m ∈ {1, . . . , n}|κ

= 1, =b

> 0} the functions ˜u

(−1)

and ˜u

(0)

have

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 55

already been deﬁned such that (5.11) holds with l replaced by m. We have

(0)

− u

(−1)

m6∈I(l)

˜p

(−1)

˜u

(−1)

m∈I(l)

˜p

(−1)

˜u

(−1)

for some 1-periodic functions ˜p

(−1)

(s) analytic on a neighbourhood of ∞ in

−1

. As before we have

˜p

(−1)

∈ A

≤−1

(−π/2, 0).(5.12)

Now we have analogues of (5.8), (5.9) and (5.10), and so the ﬁrst sum can

be written as ϕ

(−1)

+ ψ

(−1)

such that ϕ

(−1)

∈ (A

≤−1

)

−1

) and ψ

(−1)

∈

≤−1

)

(−π/2, 0) and f

(−1)

and f

(−1)

are solutions of (1.1).

Next consider the case that m ∈ I(l) and ˜p

(−1)

6= 0. Then according to

Lemma 8-3 there exist analytic 1-periodic functions p

such that ˜p

(−1)

−

+ p

and p

−

∈ A

≤−1

(α, 0) and p

∈ A

≤−1

(−π/2, β). Now deﬁne

˜u

(−1)

:= u

(−1)

+ ϕ

(−1)

m∈I(l)

−

˜u

(−1)

˜u

(0)

:= u

(0)

−

m∈I(l)

˜u

(0)

Then ˜u

(0)

−˜u

(−1)

= ψ

(−1)

m∈I(l)

(˜u

(−1)

−˜u

(0)

) ∈ (A

≤−1

)

(−π/2, 0)

and it follows that the functions ˜u

(µ)

satisfy (5.11). By induction (5.11) fol-

lows for all l. So in case N = 0 the proposition has be en proved.

Next suppose that N = 1, so β > π/2. If p

(1)

(s), m = 1, . . . , n, are the

1-periodic functions analytic on a neighbourhood of ∞ in the upper half

plane such that

(1)

− ˜u

(0)

m=1

(1)

then p

(1)

∈ A

≤−1

(0, π/2). If κ

< 1 then as before p

(1)

∈ A

≤−1

Next suppose κ

= 1 and p

(1)

6= 0. Then by Lemma 8-3 there exists an

integer N such that p

(1)

(s) = p

2πisN

(1 + o(1)) as =s → ∞ with p

6= 0

and therefore e

2πisN

(s) ∈ A

≤−1

(0, π/2). So there exists a Stokes interval

[σ

− π, σ

] of e

2πisN

which contains (0, π/2). Now σ

− π ≥ −π/2 > α

and therefore σ

> β. Hence p

(1)

∈ A

≤−1

(0, β). Moreover, if κ

= 1

then p

(1)

∈ A

≤−1

(0, π/2) according to Lemma 8. Thus, if we deﬁne

˜u

(1)

:= u

(1)

−

≤1

(1)

, then

˜u

(1)

− u

(1)

∈ (A

≤−1

)

(0, β); ˜u

(1)

− ˜u

(0)

∈ (A

≤−1

)

(0, π/2).

56 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

The fundamental systems {f

˜u

(µ)

}

l=1

, µ = −1, 0, 1, thus obtained satisfy

(5.1) and (5.2).

The uniqueness property of Proposition 10 is an immediate consequence

of a more general form of the relative Watson lemma by Malgrange and

Ramis, that can be found in [BIS]. 

6. Equations without level 1

Theorem 13 has already been stated and proven in [BF96], but here we

present a new proof.

Theorem 13. Let 0 < k

< · · · < k

= 1 be the levels in (0, 1] of Equation

(1.1), and suppose that this equation does not contain a level 1

(i.e., d

0, ∀m, l ∈ {1, . . . , n}). Let

H(s)s

Λs

G(s)

be a formal fundamental matrix

as in (1.2).

Let S

= S(α

, β

), i = 1, . . . , r, be a sequence of open sectors such that

⊃ . . . ⊃ S

, S

has aperture less than 2pπ, S

has aperture larger than

π/k

and (α

, β

) does not contain a Stokes interval of level k

, i = 1, . . . , r.

Then

H is (k

, . . . , k

)-summable on (S

, . . . , S

) with sum H

such that

(s)s

Λs

G(s)

is an analytic fundamental matrix of (1.1).

Proof. Deﬁne M

, N

, j = 1, . . . , r, to be the integers such that (M

−

1)π/2 ≤ α

< M

π/2 < N

π/2 < β

≤ (N

+ 1)π/2 and let Γ

j,µ

:= S

∩ H

µ = M

, . . . , N

, j = 1, . . . , r. Also, let S

:= C

be the Riemann surface

of s

1/p

, Γ

0,µ

:= H

, µ = M

, . . . , N

where M

:= M

, N

:= M

+ 4p − 1.

By Proposition 9 we have a representative {H

(µ)

(s)}

µ=M

of T

−1

H(s) on

the covering {H

}

µ=M

of C

such that H

(µ)

(s)s

Λs

G(s)

is an analytic

fundamental matrix of (1.1). To show that the columns

H are multi-

summable we have to construct h

l,j

∈ (A

(1/k

)

≤−k

j+1

)

), j = 0, . . . , r

such that h

l,j

j+1

≡ h

l,j+1

mod A

≤−k

j+1

, j = 0, . . . , r if k

r+1

= ∞.

Let U

(µ)

(s) := H

(µ)

(s)s

so that U

(µ)

(s)F(s) is a fundamental matrix of

(1.1) (cf. (2.2)) and let ˜u

(µ)

l,0

denote the lth column of U

(µ)

. The construction

mentioned above is equivalent to the construction of functions {˜u

(µ)

l,j

}

µ=M

for j = 1, . . . , r and l = 1, . . . , n such that:

(1) {f

˜u

(µ)

l,j

}

l=1

is a fundamental system of Equation (1.1),

(2) {˜u

(µ)

l,j

}

µ=M

represents a k

j+1

-precise quasi-function

˜u

l,j

∈ (A

(1/k

)

≤−k

j+1

)

)[log s], j = 1, . . . , r,

(3) ˜u

l,j−1

≡ ˜u

l,j

mod (A

≤−k

)

, j = 1, . . . , r.

Suppose we have constructed ˜u

l,i

for i = 0, . . . , j − 1, for some j ∈

{1, . . . , r}. Then we can apply Prop os ition 10 with α = α

, β = β

, and

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 57

with u

(µ)

= ˜u

(µ)

l,j−1

j,µ

, µ = M

, . . . , N

. Deﬁning ˜u

(µ)

l,j

:= ˜u

(µ)

, l = 1, . . . , n,

µ = M

, . . . , N

we see that properties (1), (2) and (3) are satisﬁed for i = j

as well. So they are satisﬁed for all j ∈ {1, . . . , r}.

We have ˜u

(µ+1)

l,r

− ˜u

(µ)

l,r

∈ (A

≤−1

)

(Γ

r,µ

∩ Γ

r,µ+1

). We also have ˜u

(µ+1)

l,r

−

˜u

(µ)

l,r

m=1

˜u

(µ)

m,r

for some 1-periodic analytic functions p

. Lemma 7

now tells us that each p

∈ A

≤−1

(Γ

µ,r

∩ Γ

µ+1,r

), and then it follows

from Lemma 8 and the fact that the equation has no level 1

, that p

= 0,

for all m. Hence, the functions ˜u

(µ)

l,r

, µ = M

, . . . , N

, are the restrictions of

an analytic function ˜u

l,r

∈ (A

(1/k

)

)[log s]. 

7. Equations with level 1

In this section we will consider Equation (1.1) under the assumption that

there does exist a pair (m, l) such that d

6= d

in the notation of (2.3); that

is, the equation possesses the level 1

. We will show in this section that

we can still assign a uniquely characterizable fundamental system Y(s) with

asymptotic expansion

Y(s) in appropriate regions of the Riemann surface of

the logarithm, provided <b

6= <b

if b

6= b

(cf. notation in (2.3)).

Before we state the main result of this paper (Theorem 18), we need to

deﬁne the Stokes numbers of level 1

of the equation. The Stokes number

of a function f of level 1

of the form

f(s) = exp(ds log s + 2πibs + q(s) + γ log s), with d 6= 0,(7.1)

occurring in formal solutions of equations possessing a level 1

, is associated

with curves that separate regions of growth from regions of decay of f . We

have

<{ds log s + 2πibs} = d<s log |s| − (d arg s + 2π<b)=s − 2π=b<s,

and therefore the main contribution to |f(s)| comes from exp[<{s(d log s +

2πi<b)}]. Let h be an even integer if d < 0 and an odd integer if d > 0.

Then f ∈ A

≤−1

). Moreover, f be haves as an exponential function

of order 1 in vertical strips, f becomes exponentially large on any open

sector containing H

and, if S

(h) := S((h −

)π, (h +

)π] and S

−

(h) :=

S[(h −

)π, (h +

)π) then it is easily veriﬁed that

f ∈ A

≤−1

) iﬀ ± (h + 2<b/d) < −1/2,

where the upper (lower) signs belong together.

Let {f

}

j=1

be a fundamental system of (1.1) such that u

∼ ˆu

s → ∞ on S

(h). Assume d

= d

−d

< 0 and h is even. Then there exists

∈ Z such that ±{h + 2(<b

+ N

)/d

} < −1/2. Then e

2πisN

∈

≤−1

) and therefore the solutions f

and f

+ e

2πisN

have the

same asymptotic behaviour on S

(h). So in this case it is not possible to

58 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

characterize fundamental systems Y(s) by their asymptotic behaviour

Y(s)

on S

(h).

In order for a fundamental system to be in some way uniquely determined

by its asymptotic expansion in a sector of C

∞

, which contains an open right

or left half plane, this sector therefore should contain the closure of this

half plane. However, such fundamental systems do not exist in general (see

[vdPS97, Chapter 11]).

Hence we have to characterize fundamental systems by their asymptotic

behaviour in a more complicated type of region. This region should contain

a neighbourhood of ∞ in som e half plane H

but not a neighborhood of ∞

in H

. In the case d = (−1)

h+1

a suitable region is given by

Deﬁnition 14. For θ ∈ R, h ∈ Z, we deﬁne

D(h; θ) := {s ∈ C

∞

| (h − 1)π < arg s < (h + 1)π;

(−1)

<{s(log s + iθ)} > 1}.

If h ∈ Z and θ

, θ

∈ R then D(h; θ

, θ

) := D(h; θ

) ∪ D(h; θ

We denote the boundary of D(h; θ) by C(h; θ). We have D(h; θ) ⊂

j=−1

2h+j

and se

ihπ

∈ D(h; θ) ⇔ s ∈ D(0; θ + hπ). Similarly with D

replaced by C. Details on C(0; θ) can be found in [Imm84] and [Imm91].

We have <s = O(=s/ log |s|) as s → ∞ on C(h; θ). This implies that

arg s → ±π/2 mod 2π as =s → ±∞, s ∈ C(h; θ).

If h ∈ Z and θ ∈ R, then the following properties are easily established.

• Let

θ < θ. As (−1)

<{s(log s + i

θ)} = (−1)

<{s(log s + iθ)} −

(−1)

(

θ − θ)=s, it follows that:

i) D(h;

θ) ∩ H

2h−1

⊂ D(h; θ) ∩ H

2h−1

, and that

ii) D(h;

θ) ∩ H

2h+1

⊃ D(h; θ) ∩ H

2h+1

One could say that the regions D(h; θ) ‘rotate’ (modulo some de-

formation) clockwise with increasing θ.

• D(h; θ) ∩ H

is a neighbourhood of ∞ in H

. However, since for any

∈ H

, s

+ H

6⊂ D(h; θ), the intersection D(h; θ) ∩ H

is not a

neighbourhood of ∞ in H

Let D := D(h; θ

, θ

), for some h ∈ Z and θ

< θ

. A set U ⊂ D is called

a neighbourhood of ∞ in D, if, for any θ ∈ (θ

, θ

), there exists s

∈ D

such that s

+ D(h; θ) ⊂ U.

Suppose g is an analytic function on a neighbourhood U of ∞ in D, and

suppose there exist a k > 0 and a series ˆg(s) =

n≥0

−n/p

such that for

any θ ∈ (θ

, θ

) we have



g(s) −

N−1

n=0

−n/p



≤ KA





|s|

−N/p

,(7.2)

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 59

∀s ∈ D(h; θ) ∩ U, ∀N ∈ N,

for some positive constants K and A, which only depend on θ. Then we call

g a k-Gevrey function on D (with respect to

), and the set of these

functions is denoted by A

(1/k)

(D).

Concerning 1-Gevrey functions on D we have the following theorem by

Immink (cf. [Imm96]):

Theorem 15. Suppose θ

, θ

∈ R, θ

< θ

, and h ∈ Z. Let D :=

D(h; θ

, θ

If g ∈ A

(1)

(D), then g is uniquely determined by its asymptotic expansion

ˆg.

In [Imm96] it is shown that g is already uniquely determined by its

asymptotic expansion if (7.2) with k = 1 holds on D(h; θ) for one θ ∈ (θ

, θ

Let f be a function of level 1

given by (7.1) and θ

< 2π<b/d < θ

Let h ∈ Z be odd if d > 0 and even if d < 0. Then there exist positive

constants K and c such that |f(s)| ≤ Ke

−c|s|

for s ∈ D(h; θ

) ∩ D(h; θ

) and

|s| suﬃciently large. From this it follows that (7.2) holds on this set with

g := f, g

= 0, and k = 1. This is not true if

2π<b

6∈ [θ

, θ

]. Therefore we

introduce the following deﬁnition:

Deﬁnition 16. Let f(s) be the function of level 1

given by (7.1). Then

we call

2π<b

its Stokes number. The Stokes numbers of level 1

the equation (1. 1) are the Stokes numbers of the functions e

2πijs

(s),

l, m ∈ {1, . . . , n}, j ∈ Z, that are of level 1

. That is, they are given by the

expression

2π

(<b

+ j), j, l, m as above, d

6= 0.

Suppose θ

< θ

and α < β. Let D := D(h; θ

, θ

) and S := S(α, β).

Assume D ∩ S 6= ∅. We deﬁne a neighbourhood of ∞ in D ∩ S to be an

open set U in D ∩ S, such that for any θ, γ, δ satisfying θ

< θ < θ

and

α < γ < δ < β, there exists s

∈ D ∩ S such that s

+ (D(h; θ) ∩ S(γ, δ)) ⊂

U. We write A

≤−1

(D ∩ S) for the set of functions that are analytic on a

neighbourhood of ∞ in D ∩S, and exponentially small of order 1, as s → ∞,

uniformly on D(h; θ) ∩ S(γ, δ), for any θ, γ, δ as above.

We extend Deﬁnition 2 of multisummability as follows:

Deﬁnition 17. Let 0 < k

< · · · < k

r−1

< k

= 1, and deﬁne k

r+1

= 1

Let S

⊃ . . . ⊃ S

be a nested sequence of sectors S

with aperture > π/k

i = 1 , . . . , r, aperture of S

at most 2pπ and assume S

⊃ S((h −

)π −

ε, (h+

)π+ε), for some h ∈ Z and some ε > 0. Finally, let D := D(h; θ

, θ

)

for some θ

< θ

A formal power series

f ∈ C[[s

−1/p

]]

1/(pk

)

is called (k

, . . . , k

, 1

summable on (S

, . . . , S

, D) with (k

, . . . , k

, 1

)-sum f ∈ A

(1/k

)

(D),

60 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

if there exist quasi-functions f

∈ (A

(1/k

)

≤−k

i+1

)(S

), i = 1, . . . , r, satis-

fying:

i+1

≡ f

i+1

mod A

≤−k

i+1

, i = 0, . . . , r − 1 with f

= T

−1

has a representative {f

r,ω

}

ω∈Ω

with respect

to a covering {S

r,ω

}

ω∈Ω

of S

with open sectors S

r,ω

such that f

r,ω

− f ∈ A

≤−1

(D ∩ S

r,ω

), ∀ω ∈ Ω.

Let g be another function such that f

r,ω

− g ∈ A

≤−1

(D ∩ S

r,ω

), ∀ω ∈ Ω.

Then f − g ∈ A

≤−1

(D ∩ S

r,ω

), ∀ω ∈ Ω, hence, f − g ∈ A

≤−1

(D ∩ S

) =

≤−1

(D). Theorem 15 implies f = g and it follows that f is uniquely deter-

mined by f

and D. By the relative Watson lemma ([MR92, Prop. (2.1)])

is uniquely determined by f

i−1

and S

, i = r, . . . , 1. Hence the sum f is

uniquely determined by

f and (S

, . . . , S

, D).

Similarly to Deﬁnition 2 we extend this deﬁnition to the case that

f is an

n-vector or n × n-matrix with elements in C[[s

−1/p

]]

1/(pk

)

The main result of this paper is the following theorem.

Theorem 18. Let k

< · · · < k

= 1 be the sequence of positive levels ≤ 1

of (1.1). Suppose that 1

is a level of (1.1) and that <b

6= <b

if b

6= b

where b

is deﬁned below (2.3). Let

H(s)s

Λs

G(s)

be a formal fundamental

matrix as in (1.2).

Let S

= S(α

, β

), i = 1, . . . , r, be a sequence of open sectors, such that

⊃ . . . ⊃ S

and β

− α

≤ 2pπ, β

− α

> π/k

and (α

, β

) does not

contain a Stokes interval of level k

, i = 1, . . . , r of (1.1). Moreover, suppose

that (h−1)π < α

< (h−

)π and (h+

)π < β

< (h+1)π, for some h ∈ Z.

Let

θ, θ ∈ R,

θ < θ, such that (

θ, θ) does not contain a Stokes number of

level 1

and deﬁne D = D(h;

θ, θ).

Then

H is (k

, . . . , k

, 1

)-summable on (S

, . . . , S

, D) with (k

, . . . ,

, 1

)-sum H such that (1.3) deﬁnes an analytic fundamental matrix of

(1.1).

For the proof of this theorem we will use the following lemma which

extends the results of Lemma 8-4.

Lemma 19. Let h ∈ Z and Q

be the quadrant H

∩ H

2h±1

. Here and in

the following the upper signs belong together and so do the lower signs. Let f

be given by (7.1). Deﬁne θ

2π

(<b + j), j ∈ Z, and D := D(h; θ

N−1

, θ

for some N ∈ Z.

Suppose that p(s) 6≡ 0 is a 1-periodic analytic function on a neighbourhood

of ∞ in H

2h±1

such that p(s)f(s) ∈ A

≤−1

Then there exists a 1-periodic function p

−

(s), such that p

−

(s) is analytic

on a neighbourhood of ∞ in H

2h±1

, p

(s) := p(s) − p

−

(s) is an entire

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 61

function, and

−

(s)f(s) ∈ A

≤−1

(D ∩ H

2h±1

(s)f(s) ∈ A

≤−1

(D ∩ H

2h∓1

) ∩ A

≤−1

Proof. We will give the details of the proof for the lower sign and h is even.

The other cases can be proven in a similar way.

Now Q

−

is a fourth quadrant and H := H

2h−1

is a lower half plane.

According to Lemma 8-4 we have p(s)f(s) ∈ A

≤−1

((h −

)π, hπ], and,

since p(s) 6≡ 0 also f(s) ∈ A

≤−1

((h −

)π, hπ). Therefore d < 0, f(s) ∈

≤−1

((h −

)π, (h +

)π) and θ

< θ

N−1

We have an expansion for p(s) as in (3.1) for =s < −ρ for some ρ > 0.

Let p

−

(s) :=

j≤N −1

2πisj

. Then p

−

(s) is analytic for =s < −ρ and

−

(s) = e

2πis(N−1)

O(1), s → ∞ on

H. Moreover, p

(s) =

j≥N

2πisj

an entire function.

First consider p

−

(s)f(s). The properties of p

−

imply

−

(s)f(s) ∈ A

≤−1

((h −

)π, hπ] ⊂ A

≤−1

−

).(7.3)

For ε > 0, let S

:= S((h −

)π − ε, (h −

)π + ε). In order to prove

that p

−

(s)f(s) ∈ A

≤−1

(D ∩ H) it is suﬃcient to s how, that with any θ ∈

(θ

, θ

N−1

) we c an ﬁnd positive constants K, c and ε, such that

−

(s)f(s)| ≤ Ke

−c|s|

, s ∈ D(h; θ) ∩ S

, =s < −ρ.(7.4)

As p

−

(s)f(s) = O(1) exp(ds log s + 2πis(b + N − 1) + o(s)), s → ∞ on H, it

is suﬃcient to prove that

<{ds log s + 2πis(b + N − 1)} ≤ −c|s|, ∀s ∈ D(h; θ) ∩ S

,(7.5)

for some c > 0.

So let θ ∈ (θ

, θ

N−1

). On D(h; θ) we have:

<{ds log s + 2πis(b + N − 1)}

= d<{s(log s + iθ) + is(

2π

(b + N − 1) − θ)}

< −d[=s(θ

N−1

− θ) +

2π

<s=b]

= −d sin(arg s)(θ

N−1

− θ)|s|



1 +

2π=b

d(θ

N−1

− θ)

cot(arg s)



We have d sin(arg s)(θ

N−1

−θ) > 0 on H. Furthermore, there exists an ε > 0

such that



4π=b

d(θ

N−1

−θ)

cot(arg s)



< 1, ∀s ∈ S

. Thus (7.5) and (7.4) follow.

Next consider p

(s)f(s). As p(s)f(s), p

−

(s)f(s) ∈ A

≤−1

((h −

)π, hπ],

also p

(s)f(s) belongs to this set. Moreover, f(s) ∈ A

≤−1

) and

(s)| ≤ Ke

−2πN=s

, =s ≥ −ρ for som e K > 0, and therefore p

(s)f(s) ∈

≤−1

). To prove the lemma it now suﬃc es to prove that p

f ∈

62 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

≤−1

(D ∩ H

2h+1

), hence that for any θ ∈ (θ

, θ

N−1

) we can ﬁnd positive

constants K, c and ε, such that

(s)f(s)| ≤ Ke

−c|s|

, ∀s ∈ D(h; θ) ∩ S((h +

)π − ε, (h +

)π + ε).

A proof of this inequality runs along the same lines as that of (7.4). 

The following proposition extends the results of Proposition 10.

Proposition 20. Let k

< · · · < k

be the positive levels ≤ 1 of equation

(1.1) and assume that (1.1) has a level 1

Let M be odd. Assume α < M π/2, β > (M + 2)π/2 such that I =

(α, β) does not contain a Stokes interval of level 1 of (1.1). Deﬁne Γ

∩ S(α, β) for µ = M, M + 1, M + 2. Assume

θ, θ ∈ R,

θ < θ, such

that (

θ, θ) does not contain any Stokes number of level 1

of (1.1). Deﬁne

D := D(

(M + 1);

θ, θ).

Assume we have fundamental systems {f

(µ)

}

l=1

, µ = M, M + 1, M + 2

such that for all l ∈ {1, . . . , n}:

(i) u

(µ)

∈ (A

(1/k

)

(Γ

)[log s], u

(µ)

∼ ˆu

on Γ

if µ = M, M + 1, M + 2,

(ii) u

(µ+1)

− u

(µ)

∈ (A

≤−1

)

(Γ

∩ Γ

µ+1

) if µ = M, M + 1.

Then there exists a fundamental system {f

˜u

}

l=1

of equation (1.1) such

that for all l ∈ {1, . . . , n}:

˜u

− u

(µ)

∈ (A

≤−1

)

(D ∩ H

) if µ = M, M + 2,(7.6)

˜u

− u

(M+1)

∈ (A

≤−1

)

M+1

).(7.7)

Moreover, for each l ∈ {1, . . . , n} the function ˜u

∈ (A

(1/k

)

(D)[log s]

is uniquely determined by these properties.

Remark. We can ﬁnd α and β satisfying the above conditions if and only

if [νπ − π/2, νπ + π/2] is not a Stokes interval of level 1 for any ν ∈ Z. This

corresponds to the condition that <b

6= <b

if b

6= b

Proof. Throughout the proof l, m ∈ {1, . . . , n}. Let h := (M + 1)/2, so h is

an integer. We will write m ≺ l if f

∈ A

≤−1

), and I(l) := {m | m ≺

l}. The relation ≺ gives a partial ordering on {1, . . . , n}. We will prove the

proposition by induction with respect to this partial ordering.

Let p

(µ)

, µ = M, M + 1, be the 1-periodic analytic functions such that

(µ+1)

− u

(µ)

m=1

(µ)

, µ = M, M + 1.(7.8)

As u

(µ+1)

− u

(µ)

∈ (A

≤−1

)

(Γ

∩ Γ

µ+1

) by assumption, it follows from

Lemma 7 that each of the summands must belong to this set. If m 6∈ I(l)

then e

s log s

is unbounded on H

, so <d

s is positive on H

. Therefore

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 63

6∈ A

≤−1

(Γ

∩ Γ

µ+1

) for µ = M, M + 1 and Lemma 8 tells us that

(µ)

= 0. So in (7.8) we only need to sum over m ∈ I(l).

First let l be s uch that I(l) = ∅. Then u

(µ)

is independent of µ ∈ {M, M +

1, M + 2} and ˜u

= u

(µ)

satisﬁes (7.6) and (7.7).

Now let l be such that ˜u

have been deﬁned, and satisfy (7.6) and (7.7)

for all m ∈ I(l). The functions f

˜u

, m ∈ I(l), together with the functions

(M+1)

, m 6∈ I(l), form a fundamental system of solutions. With the aid

of Lemmas 7 and 8 we may conclude as ab ove that

(M+1)

− u

(M)

m∈I(l)

˜p

(M)

˜u

for some 1-periodic analytic functions ˜p

(M)

on a neighbourhood of ∞ in

Q := Γ

∩ Γ

M+1

, and ˜p

(M)

∈ A

≤−1

(Q). According to Lemma 19

the functions ˜p

(M)

(s) can be written as p

(M)

m−

(s) + p

(M)

(s) with p

(M)

m−

∈

≤−1

(D ∩ H

) and p

(M)

∈ A

≤−1

(D ∩ H

M+2

) ∩ A

≤−1

M+1

). We

deﬁne

l,1

:= u

(M)

m∈I(l)

(M)

m−

˜u

so that u

l,1

− u

(M)

∈ (A

≤−1

)

(D ∩ H

) and

l,1

− u

(M+1)

= −

m∈I(l)

(M)

˜u

∈



≤−1



(D ∩ H

M+2

) ∩



≤−1



M+1

From these relations and assumption (ii) it follows that u

(M+2)

− u

l,1

∈

≤−1

)

(Γ

M+1

∩ Γ

M+2

) and as above we ﬁnd that there exist 1-periodic

functions p

(M+1)

m−

(s), p

(M+1)

(s), m ∈ I(l), such that

(M+2)

− u

l,1

m∈I(l)



(M+1)

m−

˜u

+ p

(M+1)

˜u



where the ﬁrst term of each summand belongs to (A

≤−1

)

(D ∩ H

M+2

), and

the second one to (A

≤−1

)

(D∩H

)∩(A

≤−1

)

M+1

). Hence, if we deﬁne

˜u

:= u

(M+2)

−

m∈I(l)

(M+1)

m−

˜u

then

˜u

= u

l,1

m∈I(l)

(M+1)

˜u

64 B.L.J. BRAAKSMA, B.F. FABER, AND G.K. IMMINK

and it is easy to verify that ˜u

satisﬁes (7.6) and (7.7). The uniqueness of

˜u

follows from Theorem 15. 

Theorem 18 can be proved similarly to Theorem 13, with the aid of Propo-

sitions 9, 10 and 20.

References

[Bal94] W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes

in Mathematics 1582, Springer Verlag, Heidelberg, 1994.

[BBRS91] W. Balser, B.L.J. Braaksma, J.-P. Ramis and Y. Sibuya, Multisummability of

formal power series solutions of linear ordinary diﬀerential equations, Asympt.

Anal., 5 (1991), 27-45.

[BT33] G.D. Birkhoﬀ and W.J. Trjitzinsky, Analytic theory of singular diﬀerence equa-

tions, Acta Math., 60 (1933), 1-89.

[Bra91] B.L.J. Braaksma, Multisummability and Stokes multipliers of linear meromor-

phic diﬀerential equations, J. Diﬀ. Eq., 92 (1991), 45-75.

[Bra92] , Multisummability of formal power series solutions of nonlinear mero-

morphic diﬀerential equations, Ann. Inst. Fourier, 42 (1992), 517-540.

[BF96] B.L.J. Braaksma and B.F. Faber, Multisummability for some classes of diﬀer-

ence equations, Ann. Inst. Fourier, 46(1) (1996), 183-217.

[BH75] B.L.J. Braaksma and W.A. Harris, Jr., On an open problem in the theory of

linear diﬀerence equations, Nieuw Archief voor Wiskunde (3), XXIII (1975),

228-240.

[BIS] B.L.J. Braaksma, G.K. Immink and Y. Sibuya, The Stokes phenomenon in

exact asymptotics, Paciﬁc J. Math., 187 (1999), 13-50.

[Eca85] J. Ecalle, Les fonctions r´esurgentes, III, Publ. Math. Orsay, 1985.

[Eca87] , L’acc´el´eration des fonctions r´esurgentes, Manuscript, 1987.

[Fab97] B.F. Faber, Diﬀeren ce equations and summability, Revista del Seminario

Iberoamericano de Matem´aticas, V (1997), 53-63.

[Imm84] G.K. Immink, Asymptotics of Analytic Diﬀerence Equations, Lecture Notes in

Mathematics, 1085, Springer Verlag, Heidelberg, 1984.

[Imm88] , Asymptotic expansions with error bounds for solutions of diﬀerence

equations of ‘level 1

’,

Equations diﬀ´erentielles dans le champ complexe, Vol. 1,

I.R.M.A., Strasbourg, (1988), 35-60.

[Imm91] , Reduction to canonical forms and the Stokes phenomenon in the theory

of linear diﬀerence equations, SIAM J. Math. Anal., 22 (1991), 238-259.

[Imm] , On the summability of the formal solutions of a class of inhomogeneous

linear diﬀerence equations, Funkcial. Ekvac., 39(3) (1996), 469-490.

[Imm96] , A particular type of summability of divergent power series, with an

application to linear diﬀerence equations, Preprint, 1996.

[Mal95] B. Malgrange, Sommation des s´eries divergentes, Expo. Math., 13 (1995), 163-

222.

[MR92] B. Malgrange and J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier,

41 (1992), 353-368.

SUMMATION METHOD FOR DIFFERENCE EQUATIONS 65

[MR91] J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability,

Ann. Inst. H. Poincar´e, Phys. Th´eor., 54 (1991), 331-401.

[RS94] J.-P. Ramis and Y. Sibuya, A new proof of multisummability of formal solutions

of non-linear meromorphic diﬀerential equations, Ann. Inst. Fourier, 44 (1994),

811-848.

[Tov96] A.I. Tovbis, Multisummable solutions of nonlinear ordinary diﬀerential equa-

tions, J. Diﬀ. Eq., 124 (1996), 359-377.

[vdPS97] M. van der Put and M.F. Singer, Galois theory of diﬀerence equations, Lecture

Notes in Mathematics, 1666, Springer Verlag, Heidelberg, 1997.

Received October 22, 1998.

University of Groningen

Department of Mathematics, P.O.Box 800

9700 AV Groningen

The Netherlands

E-mail address: B.L.J.Braaksma@math.rug.nl

University of Groningen

Department of Mathematics, P.O.Box 800

9700 AV Groningen

The Netherlands

E-mail address: bernard@math.rug.nl

University of Groningen

Department of Econometrics, P.O.Box 800

9700 AV Groningen

The Netherlands

E-mail address: G.K.Immink@eco.rug.nl

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

APPROXIMATION OF RECURRENCE IN NEGATIVELY

CURVED METRIC SPACES

Charalambos Charitos and Georgios Tsapogas

For metric spaces with curvature less than or equal to χ,

χ < 0, it is shown that a recurrent geodesic is approximated

by closed geodesics. A counter example is provided for the

converse.

1. Introduction and preliminaries.

In hyperbolic geometry it has been shown lately that many geometric prop-

erties are determined by the distance function on the space itself rather than

the diﬀerential structure. It is shown in this work that, partially, this is the

case with the notion of recurrence. For complete hyperbolic manifolds, a

recent result of Aebischer, Hong and McCullough (see [1]) states that a ge-

odesic is recurrent if and only if it is approximated by closed geodesics. We

show that, in metric spaces with curvature less than or equal to χ, χ < 0,

recurrent geodesics are approximated by closed geodesics (see Theorem 2

below). The proof of the converse statement crucially depends on the man-

ifold structure, in particular on the fact that two geodesics coincide if they

do so on an open interval. Hence, the converse statement fails in our context

due to the bifurcation property of geodesics. A counterexample exhibiting

this failure is provided in Section 4 below. A geodesic γ is called recurrent

if there exists a sequence {t

} ⊂ R, t

→ ∞ such that t

γ → γ as t

→ ∞.

Convergence in this deﬁnition is me ant to be uniform convergence on com-

pact sets which, in fact, induces the topology on the space GX consisting

of all (local) isometries R → X when X is (not) simply connected. R acts

on GX by right translations, namely, (t, g) → tg, where tg : R → X is the

geodesic deﬁned by tg (s) = g (s + t) , s ∈ R. This action is simply the ge-

odesic ﬂow. The notion of convergence in the above deﬁnition is analogous

to the tangential condition which deﬁnes recurrence in the manifold case.

We use the notion of approximation given in Deﬁnition 6 below which was

introduced in [1] in order to characterize recurrent geodesics in hyperbolic

manifolds.

X will always denote a locally compact, complete, geodesic m etric space

with curvature less than or equal to χ, χ < 0. Recall that a geodesic metric

space is said to have curvature less than or equal to χ if each x ∈ X has

a neighborhood V

such that every geo des ic triangle of perimeter strictly

68 C. CHARITOS AND G. TSAPOGAS

less than

2π

√

(=+∞ when χ ≤ 0) contained in V

satisﬁes CAT − (χ)

inequality (see [11] for deﬁnitions and basic properties). We will denote

the metric by d (·, ·) and will use the same letter to denote distance when

the metric space to which we refer is understood. All curves are assumed

to be parametrized by arclength. A geodesic segment in X is an isometry

c : I → X, where I is a closed interval in R. A geodesic in X is a map

c : R → X such that for each closed interval I ⊂ R, the map c |

: I → X is

a geodesic segment. A local geodesic segment (usually called geodesic arc)

in X is a map c : I → X such that for each t ∈ I there is an ε > 0 such

that c |

[t−ε,t+ε]∩I

: [t − ε, t + ε] ∩ I → X is a geodesic segment. Similarly, a

local geodesic R → X is deﬁned. A closed geodesic in X is a local geodesic

c : R → X which is a periodic map.

Deﬁnition 1. An oriented geodesic g in X is said to be approximated by

closed geodesics if, for every ε > 0 and every x ∈ Im g, there exists a closed

oriented geodesic c such that for some point y ∈Im c,

d (c (t + t

) , g (t + t

)) < ε

for all t ∈ [0, period (c)] , where t

, t

∈ R with x = g (t

) and y = c (t

) .

The following theorem is the main result of this paper.

Theorem 2. Let X be a locally compact, complete, geodesic metric space

which has curvature less than or equal to χ, χ < 0. If a geodesic or geodesic

ray in X is recurrent, then it is approximated by closed geodesics.

The proof of Theorem 2 uses the notion of quasi-geodesic and its stability

properties. We will closely follow notation and terminology app earing in [8,

Ch. 3] where we refer the reader for ﬁrst deﬁnitions and basic prope rties of

quasi-geodesics. Here we only recall the following deﬁnition.

Deﬁnition 3. Let f : [a, b] → X be a continuous map with −∞ ≤ a ≤ b ≤

+∞ and λ, κ, L real numbers with λ ≥ 1, κ ≥ 0, L > 0.

f is a (λ, κ, L) −quasi-geodesic if for every subinterval [a

, b

] of [a, b] sat-

isfying

length f



, b



≤ L,

the following inequality holds

length f



, b



≤ λd





, f





+ κ.

The next proposition is a well know fact for CAT −(χ) spaces. We include

a short proof of it, since it is diﬃcult to ﬁnd exact reference (when X is a

geometric polyhedron this result follows from [3, p. 403]).

Proposition 4. Let M be a complete geodesic space satisfying CAT − (χ)

inequality with χ < 0. Every local geodesic segment in M is a geodesic seg-

ment.

RECURRENCE IN NEGATIVELY CURVED SPACES 69

Proof. Let δ : [0, L] → M be a local geodesic segment in M, L > 0. Set

l = sup



t ∈ [0, L]



δ |

[0,t]

is a geodesic segment



Apparently, l > 0 and by completeness of M, δ |

[0,l ]

is a geodesic segment

joining δ (0) with δ (l). Assuming the conclusion is not true, i.e., l < L, let ε

be a p ositive number such that δ |

[l −ε,l+ε]

is a geodesic segment. Denote by

[δ (0) , δ (l + ε)] the geodesic segment in M joining δ (0) with δ (l + ε). Since

δ |

[0,l +ε]

is not the geodesic segment joining δ (0) with δ (l + ε) ,

d (δ (0) , δ (l + ε)) < d (δ (0) , δ (l)) + d (δ (l) , δ (l + ε)) .(1)

The points δ (0) , δ (l ) and δ (l + ε) deﬁne a geodesic triangle in M. Denote

by ∆ =



δ (0), δ (l ), δ (l + ε)



the corresponding comparison triangle which

is non-degenerate by inequality (1). Choose points B on δ |

[0,l ]

and B

δ |

[l ,l+ε]

such that d (B, δ (l )) = d (B

, δ (l)) = ε

< ε and denote by B and

the corresponding points on the comparison triangle. Then by (1) the

angle of ∆ at δ (l) is smaller than π and therefore



B, B



< d



B, δ (l)



+ d



δ (l), B



= 2ε

By comparison, d (B, B

) ≤ d



B, B



so we obtain



B, B



< d (B, δ (l)) + d



δ (l) , B



This contradicts the fact that δ |

[l −ε

,l +ε

]

is a geodesic segment. 

Let

X be the universal cover of X and p :

X → X the projection map.

X becomes a metric space as follows: Given ex, ey ∈

X choose any curve

ec : [a, b] →

X with ec (a) = ex and ec (b) = ey and deﬁne the distance from ex to

ey to be the length of the unique length minimizing curve in the homotopy

class of pec with endpoints ﬁxed. For the existence of the length minimizing

curve see [10]. This distance function is a metric on

X which inherits the

properties of X, namely,

X becomes a complete geodesic locally compact

(hence, proper) me tric space. π

(X) acts on

X and the action commutes

with p. As the projection p is a local isometry, it follows that π

(X) acts on

X by local isometries. Using the fact that

X is geodesic and Proposition 4,

it is routine to show that π

(X) acts on

X by isometries. In addition,

has curvature less than or equal to χ, χ < 0 and, by a theorem of Gromov

(see for example [11, p. 325]),

X satisﬁes CAT − (χ) inequality.

GX is by deﬁnition the space of all local geodesics R → X and, by Propo-

sition 4 above, G

X is the space consisting of all global geodesics R →

The topology on these spaces is uniform convergence on compact sets. The

boundary ∂

X can be deﬁned using either equivalence classes of sequences

or, equivalence classes of geodesic rays. The local compactness assumption

on X implies that

X is proper and hence the two deﬁnitions coincide (see

70 C. CHARITOS AND G. TSAPOGAS

[8, Ch. 2]). We will be using them interchangeably. For any two distinct

points ξ, η in ∂

X there exists a unique, up to parametrization, (oriented)

geodesic g with g (−∞) = ξ and g (∞) = η (se e for example [5, Prop. 2]).

We need the following lem ma which asserts that the projection of a point

onto a geodesic always exists.

Lemma 5. Let g be a geodesic in G

X (or a geodesic segment) and x

point in

X. There exists a unique real number s such that g (s) realizes the

distance of x

from Im g, i.e., dist (x

, Im g) = d (x

, g (s)) .

Proof. We may assume that x

/∈Im g. Existence is apparent. Assume that

s 6= s

are two such numbers. The points g (s) , g (s

) and x

deﬁne a non-

degenerate geodesic triangle in

X and denote by ∆ =



g (s), g (s

), x



the

corresponding comparison triangle. ∆ is an equilateral triangle in the unique

complete simply connected Riemannian 2-manifold of constant sectional cur-

vature χ. Hence, the angles of ∆ at g (s) and g (s

) are each less than π/2.

Therefore, there exists a point g (t) on the side of ∆ opposite to x

such

that d



, g (t)



< d



, g (s)



= d



, g (s

)



. By CAT − (χ) inequality,

d (x

, g (t)) ≤ d (x

, g (s)) , a contradiction. 

Remark 1. If c ∈ GX is a closed geodesic and x

∈ X, the same argument

applied to a lifting ec of c shows that there exists a unique point B ∈Im c

such that d (x

, B) = dist (x

, Im c) .

Remark 2. Set ∂

X =

(ξ, η) ∈ ∂

X × ∂

X : ξ 6= η

and let ρ : G

X → ∂

be the ﬁber bundle given by ρ (g) = (g (−∞) , g (+∞)). Since for any two

distinct points ξ, η in ∂

X there exists a unique (oriente d) geodesic g with

g (−∞) = ξ and g (∞) = η (see for example [5, Prop. 2]), the ﬁber of ρ is

R. Moreover, this bundle is trivial (see for example [4, Th. 4.8]). To deﬁne

a trivialization, let x

be a base point and let

H : G

≈

−→ ∂

X × R(2)

be the trivialization of ρ with respect to x

deﬁned by

H (g) = (g (−∞) , g (+∞) , s)

where −s is the real number provided by Lemma 5. Note that the composite

of the geodesic ﬂow R × G

X → G

X with H is given by the formula

(ξ

, ξ

, s) −→ (ξ

, ξ

, s + t)

for all (ξ

, ξ

) ∈ ∂

X and s ∈ R.

RECURRENCE IN NEGATIVELY CURVED SPACES 71

2. Recurrent geodesics.

Deﬁnition 6. A geodesic γ in X is called recurrent if there exists a sequence

} ⊂ R, t

→ ∞ such that t

γ → γ as t

→ ∞.

For a recurrent geodesic γ in X there exists a sequence of closed (in

fact, piece-wise geodesic) curves {γ

}

n∈N

, associated to γ as follows: Fix

a convex neighborhood U of γ (0) , i.e., a neighborhood which satisﬁes the

following property: For all x, y ∈ U there exists a unique geodesic segment

with endpoints x and y lying entirely in U. Such a neighborhood exists (see

for example [2]). If {t

} is the sequence given by Deﬁnition 6 above and

= d (γ (0) , γ (t

)), let K ∈ N such that γ (t

) ∈ U for all n ≥ K. Deﬁne

, n ≥ K to be the curve

: [0, t

+ ε

] → X(3)

with γ

(t) = γ (t) ∀ t ∈ [0, t

] and γ

+ε

]

the unique geodes ic segment

in U joining γ (t

) with γ (0) . Note that t

+ ε

is the period of the closed

curve γ

. In the sequel, we will refer to these closed curves by writing

, n ∈ N but it will always b e implicit that n is large enough so that γ

are

deﬁned.

Using the following lemma, we may assume that given a recurrent geodesic

γ, the associated closed curves {γ

}

n∈N

are not homotopic to a point.

Lemma 7. Given a recurren t geodesic γ there exists M ∈ N such that each

closed curve γ

, n ∈ N associated to γ is not homotopic to a point, provided

n ≥ M.

Proof. Let eγ be a lift of γ to the universal cover

X of X parametrized so that

eγ (0) projects to γ (0) = γ

(0) . The curve γ

[0,t

]

is a local geodesic segment

and, by Proposition 4, its lift fγ

[0,t

]

X starting at eγ (0) is a geodesic

segment. Moreover, γ

+ε

]

and its lift fγ

+ε

]

X starting at

eγ

) are both geodesic segments. We have

d (eγ

+ ε

) , eγ

(0)) ≥ d (eγ

) , eγ

(0)) − d (eγ

+ ε

) , eγ

))

= t

− ε

Since ε

→ 0 and t

→ ∞ as n → ∞, we may choose M ∈ N such that

eγ

+ ε

) , eγ

(0) are distinct for all n ≥ M. Therefore, fγ

[0,t

+ε

]

, which

is the lift of the closed curve γ

starting at eγ (0) = fγ

(0) , has distinct

endpoints and, therefore, γ

, n ≥ M is not homotopic to a point. 

The following proposition shows that the lifts (to the universal cover

of the closed curves γ

associated to a recurrent geodesic γ are, for n large

enough, quasi-geodesics with arbitrarily large L. Recall that a CAT − (χ)

space is a δ−hyperbolic space in the sense of Gromov (see for example [11,

72 C. CHARITOS AND G. TSAPOGAS

Sec. 2]). This applies to the universal covering

X, since it satisﬁes CAT −(χ)

inequality globally. Let δ denote the hyperbolicity constant of the space

Proposition 8. Let γ be a recurrent geodesic in X and {γ

}

n∈N

the as-

sociated closed curves. For every L > 0, there exists N ∈ N such that all

lifts fγ

: R →

X of γ

with n ≥ N are (λ, κ, L) −quasi-geodesics provided

κ > 16δ and λ = 1, where δ is the h yperbolicity const ant of

Proof. Let γ be a recurrent geodesic and L > 0 be given. The sequence {t

}

given by Deﬁnition 6 converges to inﬁnity. Moreover, ε

= d (γ (0) , γ (t

)) →

0 and t

+ ε

= period (γ

) also converges to inﬁnity as n → ∞. Hence, we

may choose N such that

+ ε

> L and ε

(κ − 16δ) for all n ≥ N.(4)

Let now [a, b] be any interval with b − a < L (cf. Deﬁnition 3). For each

n ≥ N there exists an integer k

such that

fγ

([a, b]) ⊂ fγ

([(k

− 1) (t

+ ε

) , k

+ ε

) + t

]) .(5)

Denote by [fγ

(a) , fγ

(b)] the unique geodesic segment in

X joining fγ

(a)

with fγ

(b) and set

:= fγ

+ ε

) − ε

)

:= fγ

+ ε

)) .

The distance of any point on [fγ

(a) , fγ

(b)] from fγ

([a, b]) is bounded by a

number which depends on the hyperbolicity constant δ of the space

Xand on

the number of geodesic segments which constitute fγ

([a, b]) , see [8, Lemma

1.5, p. 25]. In our case here, fγ

([a, b]) consists of at most three geodesic seg-

ments (since the right hand side of inclusion (5) above consists of 3 geodesic

segments) and the bound is 8δ. Hence we have

d (y

, [fγ

(a) , fγ

(b)]) ≤ 8δ.(6)

By Lemma 5, let B

be the point on [fγ

(a) , fγ

(b)] which realizes the dis-

tance in the left hand side of inequality 6. Assume that neither fγ

(a) nor

fγ

(b) lies on the geodesic segment [A

, y

] . Then we have the following

triangle inequalities

d (fγ

(a) , A

) ≤ d (fγ

(a) , B

) + d (B

, y

) + d (y

, A

)

d (y

, fγ

(b)) ≤ d (y

, B

) + d (B

, fγ

(b))

which, after employing the fact that d (A

, y

) = ε

, become

length fγ

([a, b]) = d (fγ

(a) , A

) + d (A

, y

) + d (y

, fγ

(b))

≤ 2ε

+ 2d (y

, B

) + d (fγ

(a) , fγ

(b))

by inequality (6)

≤ 2ε

+ 2 · 8δ + d (fγ

(a) , fγ

(b))

by inequality (4)

≤ κ + d ( fγ

(a) , fγ

(b)) .

RECURRENCE IN NEGATIVELY CURVED SPACES 73

The case where fγ

(a) and/or fγ

(b) lies on [A

, y

] is treated similarly. 

Corollary 9. For n ∈ N suﬃciently large, the isometry of

X in π

(X)

which corresponds to the homotopy class of the closed curve γ

is hyperbolic.

Proof. It suﬃces to show that each fγ

: R →

X determines exactly two

boundary points fγ

(−∞), fγ

(+∞) . By Lemma 8 there exists an M ∈ N

such that fγ

is a quasi-isometry for all n ≥ M. Each such fγ

induces a

map ∂R → ∂

X which is a homeomorphism onto its image, see [8, Th. 2.2,

p. 35]. As ∂R consists of two distinct points, fγ

(−∞), fγ

(+∞) ∈ ∂

X are

also distinct for all n ≥ M. 

It now follows that a recurrent geodesic γ in X as well as each of the

(oriented) closed curves γ

, n ≥ M (cf. Lemma 7 and Corollary 9 above)

determine exactly two boundary points in ∂

X denoted by eγ (−∞), eγ (+∞)

and fγ

(−∞), fγ

(+∞) respectively. We need the following lemma concern-

ing these boundary points. Recall that

X ∪ ∂

X is a compact space which is

metrizable (see [8, p. 134]), and we will denote such metric by d

X∪∂

Lemma 10. fγ

(−∞) → eγ (−∞) and fγ

(+∞) → eγ (+∞) as n → ∞.

Proof. As above, let ε

= length (Im γ

) − t

so that t

+ ε

is the pe-

riod of γ

. We ﬁrst show that fγ

(+∞) → eγ (+∞) . Consider the sequence

fγ

(k (t

+ ε

)) , k ∈ N which converges to fγ

(+∞) as k → ∞. Thus, there

exists k

∈ N such that

X∪∂

(fγ

+ ε

)) , fγ

(+∞)) < 1/n.(7)

Now consider the sequences y

:= fγ

+ ε

)) and x

:= eγ (t

), n ∈ N.

Since x

→ eγ (+∞), by inequality (7) above it is enough to show that the

sequences {x

} and {y

} represent the same element in ∂

X or, in other

words, that the hyperbolic product (x

, y

)

with respect to the base point

:= eγ (0) converges to +∞ as n → +∞. For the notion of hyperbolic

product of sequences and their equivalence, see [8].

The stability property of quasi-geodesics states (see Corollary 1.10 of [8,

p. 31]) that given any two numbers κ ≥ 0 and λ ≥ 1, there exists a constant

C depending on λ, κ and on the hyperbolicity constant δ of the space such

that if L is bigger than 2C then every (λ, κ, L)-quasi-geodesic f : [a, b] →

lies within a C−neighborhood of the geodesic segment [f (a) , f (b)] . By

choosing λ = 1, κ > 16δ where δ is the hyperbolicity constant of the space

X and L > 2C we obtain, by Proposition 8 above, a natural number N

such that all fγ

: R →

X with n ≥ N are (λ, κ, L)-quasi-geodesics. In

particular, fγ

: [0, k

+ ε

)] →

X are (λ, κ, L)-quasi-geodesics for all

n ≥ N. Therefore, by Corollary 1.10 of [8, p. 31] as explained above,



, x



< C ∀ n ≥ N

74 C. CHARITOS AND G. TSAPOGAS

where x

denotes the projection of x

on the geodesic segment [eγ (0) , y

]

(cf. Lemma 5). Hence,

, y

)

(d (x

, x

) + d (y

, x

) − d (x

, y

))

≥

(d (x

, x

) − C + d (y

, x

) − d (x

, y

) − C)

= (x

, y

)

− C

= d (x

, x

) − C.

Apparently, d (x

, x

) → ∞ as n → +∞ and, hence, (x

, y

)

→ ∞ as

required.

In order to show that fγ

(−∞) → eγ (−∞) we work in a similar manner:

The se quence fγ

(−k (t

+ ε

)) , k ∈ N converges to fγ

(−∞) as k → ∞.

Hence, there exists k

∈ N such that d

X∪∂



fγ

(−k

+ ε

)) ,

fγ

(−∞)



< 1/n. As before, sequences {y

} and {x

} are deﬁned by y

fγ

(−k

+ ε

)) and x

:= eγ (−t

), n ∈ N. Then we use the same argu-

ments to show that the hyperbolic product (x

, y

)

with respect to the

base point x

:= eγ (0) converges to +∞ as n → +∞. 

3. Proof of main theorem.

Let γ be a recurrent geodesic, ε > 0 and x ∈ Im γ be given. We may assume

that x = γ (0) . Let {t

} be the sequence given by Deﬁnition 6 and {γ

} the

sequence of the associated closed curves given by formula (3) above. For

each n ∈ N, there exists a unique closed geodesic c

in the free homotopy

class of γ

. The number t

+ ε

is the period of γ

and let s

denote the

period of c

(apparently, s

< t

+ ε

). Let B

be the projection of γ (0)

onto Im c

, i.e., d (γ (0) , B

) = d (γ (0) , Im c

) . Such a point exists and is

unique by Remark 1 following Lemma 5. Lift γ to an isome try eγ : R →

with a base point eγ (0) satisfying p (eγ (0)) = γ (0) , where p :

X → X is

the universal covering map. Lift each c

to an isometry ec

: R →

X and

parametrize it so that ec

(0) is a point

satisfying



, eγ (0)



= d (B

, γ (0)) and p





= B

For the reader’s convenience, we have gathered all the above notation in

Figure 1.

Since

p (fγ

+ ε

)) = p (eγ (0)) = γ (0)

and γ

, c

are homotopic, the isometry φ

X which translates ec

(in the

positive direction) satisﬁes

(eγ (0)) = fγ

+ ε

) .

RECURRENCE IN NEGATIVELY CURVED SPACES 75

s s ss

s s s s

eγ

eγ(0)

(0) =

eγ(s) eγ(t

) eγ(t

+ 

)

fγ

+ 

)

(s)

) = φ





Figure 1.

Moreover,



fγ

+ ε

) , φ





= d



(eγ (0)) , φ





= d



eγ (0) ,



(8)

We now proceed to show that given ε > 0, there exists N ∈ N such that for

all n ≥ N

d (eγ (s) , ec

(s)) < ε ∀ s ∈ [0, s

] .(9)

Recall that s

is the period of c

and s

< t

+ ε

= period (γ

) . Using

Lemma 10 and the fact that γ

, c

are homotopic for all n large enough, we

have that ec

(+∞) → eγ (+∞) and ec

(−∞) → eγ (−∞) . Let H : G

≈

−→

∂

X ×R be the trivilization of the ﬁber bundle G

X −→ ∂

X with respect to

the base point x

= eγ (0) . This homeomorphism was described in Remark

2 following Lemma 5. By the choice of parametrization for each ec

made

above



i.e., ec

(0) =



, we have that H

−1

( ec

(−∞) , ec

(+∞) , 0) = ec

Moreover, H

−1

(eγ (−∞) , eγ (+∞) , 0) = eγ and, thus, ec

→ eγ uniformly on

compact sets. Observe that such convergence is weaker than property (9).

However, it implies, in particular, that dist (eγ (0) , Im ec

) → 0 as n → ∞.

Hence, we may choose N ∈ N such that



eγ (0) ,



< ε/5, for all n ≥ N.(10)

Moreover, we may choose N such that, in addition, the following inequality

is satisﬁed

= d (eγ (t

) , eγ (t

+ ε

)) < ε/5, for all n ≥ N.(11)

To show inequality (9), let s ∈ [0, s

] be arbitrary and let D

(resp. F

) be the point on the geodesic segment [eγ (0) , fγ

+ ε

)]

76 C. CHARITOS AND G. TSAPOGAS



resp.

eγ (0) , φ



i

whose distance from eγ (0) is equal to s. Then,

d (eγ (s) , ec

(s)) ≤ d (eγ (s) , D

) + d (D

, F

) + d (F

, ec

(s))

≤ d (eγ (s) , D

) + d (D

, F

) + d



, F



+ d



, ec

(s)



where F

is the point on

, φ



i

satisfying



, φ





= d



, φ





By comparison (see for example [12, Prop. 29]) we have

d (eγ (s) , D

) ≤ d (eγ (t

+ ε

) , fγ

+ ε

)) ≤ 2ε

d (D

, F

) ≤ d



eγ (t

+ ε

) , φ







, F



≤ d



eγ (0) ,





, ec

(s)



≤





eγ (0) , φ





− d



, φ







< d



eγ (0) ,



Combining the above inequalities with inequalities (8), (10) and (11), we

obtain property (9) which completes the proof of the existence of a sequence

of closed geodesics approximating a given recurrent geodesic. 

Remark. Let Γ be a discrete group of isometries of a locally compact, com-

plete geodesic metric space Y satisfying CAT − (χ) inequality, χ < 0. The

notion of controlled concentration points in the limit set of Γ can be deﬁned

as follows. ξ ∈ ∂Γ is a controlled concentration point if it admits a neighbor-

hood U containing ξ with the following property: For every neighborhood

V of ξ there exists an element γ ∈ Γ such that γ (U) ⊂ V and ξ ∈ γ (V ) .

Following [1], one can show that ξ is a controlled concentration point if and

only if there exists a sequence of {φ

} of distinct elements of Γ such that

(ξ) → ξ and φ

(0) → η with η 6= ξ. The proof in this more general

setting is identical with the one provided in [1] except that the convergence

property used there, namely, φ

(x) → η for all x ∈ Y ∪ ∂Y, is provided

in our case by Proposition 7.2 in [6, Ch. 1]. The latter property for ξ is

equivalent to the existence of a recurrent geodesic γ with γ (+∞) = ξ and

γ (−∞) = η. Hence we obtain the following connection between recurrent

geodesics and controlled concentration points which also holds for manifolds

(see [1]).

Theorem 11. Let Y be a locally compact, complete geodesic metric space Y

satisfying CAT − (χ) inequality, χ < 0 and Γ a discrete group of isometries

of Y. A limit point ξ ∈ ∂Y is a controlled concentration point if and only if

γ (+∞) = ξ for some recurrent geodesic γ in Y.

RECURRENCE IN NEGATIVELY CURVED SPACES 77

4. Construction of a counterexample.

As it was mentioned in the introduction, approximation by closed geodesics

does not imply recurrence. The following example demonstrates the exis-

tence of a geodesic in a CAT − (χ) , χ < 0 space which is not recurrent but

can be approximated by closed geodesics in the sense of Deﬁnition 1. Let X

be the union of two hyperbolic cylinders identiﬁed along a (convex) geodesic

strip bounded by two geodesic segments (see Figure 2). We may adjust the

geometry of X so that the unique simple closed geodesic in each cylinder,

denoted by c

and c

, have a common image in the geodesic strip, namely,

the geodesic segment indicated by letters A and B in Figure 2. Using Cor. 5

of [2] and the fact that the geodesic strip is a convex closed subset it follows

that X is a CAT − (χ) space with χ < 0.

Figure 2.

Let ω

and ω

be the periods of c

and c

respectively and assume that

and c

are parametrized so that c

(0) = c

(0) = B and clockwise i.e.,

(s) = c

(s) for all s ∈ [0, d (A, B)] . Deﬁne γ : R → X as follows:

γ (t) = c

(t) , for t ∈ [0, ω

]

γ (t) = c

(t) , for t ∈ (−∞, 0] ∪ [ω

, +∞) .

It is apparent that γ can be approximated by closed geodesics in the sense

of Deﬁnition 1. We pro cee d to show that γ is not recurrent by showing that,

γ and sγ are not close in the compact open topology for any positive real

s. For this it suﬃce s to show that there e xists ε > 0 and a compact M ⊂ R

78 C. CHARITOS AND G. TSAPOGAS

such that for any positive s ∈ R,



sγ (t

) , γ (t

)



≥ ε for some t

∈ M.(12)

For simplicity, we may assume that d (A, B) = ω

/2 = ω

/4. Pick ε <

d (A, B) /2 and cho os e a compact M ⊂ R containing the real numbers 0 and

3ω

/4. Let s be arbitrary positive real. If



γ (s) , γ (0)



= d



sγ (0) , γ (0)



≥ ε

then Equation (12) is satisﬁed for the number t

= 0. If d



γ (s) , γ (0)



< ε

then for t

= 3ω

/4 we have



sγ (t

) , γ (t

)



= d



sγ



3ω



, c



3ω



d (A, B)

> ε.

This completes the proof that γ is not recurrent and, therefore, approxima-

tion by closed geodesics does not imply recurrence.

References

[1] B. Aebicher, S. Hong and McCullough, Recurrent geodesics and controlled concentra-

tion points, Duke Math. J., 75(3) (1994), 759-774.

[2] W. Ballman, Singular spaces of non-positive curvature, in Sur les groups hyper-

boliques d’apr´es Gromov (Seminaire de Berne), ´edit´e par E. Ghys et P. de la Harpe,

(a paraitre chez Birkh¨auser), 1990.

[3] M.R. Bridson, Geodesics and geometry in metric simplicial complexes, in Group The-

ory from a Geometrical Viewpoint, (ICTP, Trieste, Italy, March 26 - April 6, 1990),

E. Ghys and A. Haeﬂiger eds. (1991).

[4] Ch. Champetier, Petite simpliﬁcation dans les groupes hyperboliques, Ann. Fac. Sci.

Toulouse, VI. Ser., Math., 3(2) (1994), 161-221.

[5] C. Charitos, Closed geodesics in ideal polyhedra of dimension 2, Rocky Mountain

Journal of Mathematics, 26(1) (1996), 507-521.

[6] M. Coornaert, Sur les groupes proprement discontinus d’isom´e tries des espaces hy-

perboliques au sens de Gromov, Th`ese U.L.P., Publication de l’IRMA.

[7] , Measures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens

de Gromov, Paciﬁc J. Math., 159(2) (1993), 241-270.

[8] M. Coornaert, T. Delzant and A. Papadopoulos, G´eom´etrie et th´eorie des groupes,

Lecture Notes in Mathematics, 1441, Springer-Verlag, (1990).

[9] M. Gromov, Hyperbolic groups, in ‘Essays in Group Theory’, MSRI Publ., 8, Springer

Verlag, (1987), 75-263.

[10] , Structures m´etriques pour les vari´et´es riemanniennes, written with J. La-

fontaine and P. Pansu, Cedic Fernand Nathan, Paris, 1981.

[11] F. Paulin, Constructions of hyperbolic groups via hyperbolization of polyhedra, in

Group Theory from a Geometrical Viewp oint, (ICTP, Trieste, Italy, March 26 - April

6, 1990), E. Ghys and A. Haeﬂiger eds. (1991).

RECURRENCE IN NEGATIVELY CURVED SPACES 79

[12] M. Troyanov, Espaces `a courbure n´egative et group´es hyperboliques in Sur les groups

hyperboliques d’apr`es Gromov (Seminaire de Berne), ´edit´e par E. Ghys et P. de la

Harp e, (a paraitre chez Birkh¨auser), 1990.

Received September 23, 1998 and revised May 27, 1999.

Agricultural University of Athens

75 Iera Odos

Athens 11855

Greece

E-mail address: bakis@auadec.aua.gr

University of t he Aegean

Karlovassi

Samos 83200

Greece

E-mail address: gtsap@aegean.gr

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

COHOMOLOGY OF SINGULAR HYPERSURFACES

Bernard M. Dwork

Professor Dwork passed away on May 9, 1998 after a long

illness. The manuscript was completed a few days earlier, and

was submitted to the Paciﬁc Journal of Mathematics follow-

ing his express desire. It is a testimony to his dedication to

mathematics even during his last ill ness - Managing Editor.

Part I.

Our object is to extend earlier work [D1] on singular hypersurfaces deﬁned

over an algebraic number ﬁeld to singular hypersurfaces deﬁned over function

ﬁelds in characteristic zero.

A key role will be played by the results of Bertolin [B1] which in turn is

based upon the Transfer Theorem of Andr´e–Baldassarri–Chiarellotto [DGS,

Theorem VI 3.2].

Let h(A, x) be the generic form of degree d in n+1 variables x

, . . . , x

n+1

Thus letting F



u ∈ N

n+1



n+1

i=1

= d



h(A, x) =

u∈F

where the symbols {A

}

u∈F

are algebraically independent over Q.

Let E

= x

∂

∂x

(1 ≤ i ≤ n + 1), h

= E

h. Let R(A) be the resultant of

, h

, . . . , h

, h}.

Let V be an absolutely irreducible subvariety of the discriminant locus,

R(A) = 0. Let k be the ﬁeld of deﬁnition of V .

Let Ω be a suitable universal domain in characteristic zero, and let L

∗

Ω

the ring of all formal sums

∗

Ω

(

∗

u∈F



∈ π

Ω

)

where F = {u = (u

, u

, . . . , u

n+1

) | du

= u

+ · · · + u

n+1

} and where

p−1

= −p, p a rational prime. (Thus π need not be in Ω.)

For λ ∈ V , λ rational over Ω we write

∗

i,λ

= γ

−

◦ (E

+ πx

(λ, x)) 1 ≤ i ≤ n + 1

82 BERNARD M. DWORK

an endomorphism of L

∗

Ω

where γ

−

is the projection operator

−

(

0 if any v

≥ 1

if all v

≤ 0.

For each integer , let K

(`)

be the set of all ξ

∗

∈ L

∗

Ω

such that ξ

∗

is annihilated

by all monomials of degree  in {D

∗

i,λ

}

1≤i≤n+1

In the following, ord refers to a rank one valuation of Ω.

For b ∈ R, b > 0, let L

∗

(b) =

u∈F



inf

(ord C

+ u

b) > −∞

Let Γ be an indeterminate and consider the polynomial h(λ, x)+Γh(A, x).

Let R(λ, Γ, A) be the resultant of

(h(λ, x) + Γh(A, x)), . . . , E

n+1

(h(λ, x) + Γh(A, x))

and write

R(λ, Γ, A) = Γ

(ρ

(λ, A) + Γρ

(λ, A) + Γ

(λ, A) + · · · ),

where ρ

(λ, A) 6= 0. The key result of the research of Bertolin [B1, Theorem

3.11] states that:

Theorem 1.

(`)

⊂ L

∗

(τ(n, d, e, ) ord ρ

(λ, A) + ε)

for all ε > 0. Here τ (n, d, e, ) depends only on n, d, e and  and is indepen-

dent of the coeﬃcients of h(λ, X).

Remark. Be rtolin obtains estimates independent of . The estimate given

here depends upon  but is simpler to state. The slight error in [B1, Theorem

3.11] is corrected in [B2].

Corollary 1. If λ ∈ V and ρ

(λ, A) 6= 0, then K

(`)

⊂ L

∗

(ε) for all  and

all ε > 0 and for all but a ﬁnite set of valuations (depending on λ).

Corollary 2. For λ ∈ V with ρ

(λ, A) 6= 0, dim K

(`)

is independent of λ.

Proof. We choose a valuation v of k(λ) such that (extending the valuation

of k(λ) to k(λ, A) via the Gauss norm relative to A)

|ρ

(λ, A)|

= 1

|λ|

≤ 1.

By the Lemma of Appendix B, we may choos e a generic point λ

of V over

k so close to λ v-adically that |λ − λ

< 1 and hence |ρ

(λ

, A)|

= 1. Thus

(`)

and K

(`)

lie in L

∗

(ε) (v-adically) for all ε > 0 and hence T

λ,λ

= γ

−

◦exp

πX

(h(λ

, x) − h(λ, x)) is an isomorphism between K

(`)

and K

(`)

as vector

spaces over Ω. 

COHOMOLOGY OF SINGULAR HYPERSURFACES 83

Part II: Koszul compl ex.

In earlier work [D1, Theorem 19.2] we discussed the (cohomological) Koszul

complex of D

∗

1,λ

(0)

, . . . , D

∗

n+1,λ

(0)

operating on K

(∞)

(0)

∞

[

`=1

(`)

(0)

where λ

(0)

is algebraic over Q. We denote by H

(s)

(∞)

(0)

) the s-th cohomology group

of this complex. We showed:

Theorem 2.

dim H

(s)



(∞)

(0)



< ∞.

We also showed [D1, Theorem 17.1] that this dimension can be bounded

in terms of d and n alone.

Note. Equation 19.4 of [D1] is stated without proof. This gap will b e

ﬁlled in Appendix A.

Corollary 3. For λ

∈ V, dim H

(s)

(∞)

) < ∞ and if ρ

(λ

, A) 6= 0, then

dim H

(s)

(∞)

) is independent of λ

Proof. We choose λ

(0)

algebraic over Q such that λ

(0)

∈ V and ρ

(λ

(0)

, A) 6=

0. We choose a valuation v such that





(0)

, A





= 1,



(0)



≤ 1

and then choose a generic point λ of V ove r k in Ω as in the proof of Corollary

2. Then T

(0)

,λ

provides an isomorphism of K

(s)

(0)

with K

(s)

which induces an

isomorphism of H

(s)

(∞)

(0)

) with H

(s)

(∞)

) for all s. This shows ﬁniteness

for λ generic.

If ρ

(λ

, A) 6= 0, then by the same argument choosing λ generic close to

we c onclude that dim H

(s)

(∞)

) = dim H

(s)

(∞)

). If ρ

(λ

, A) = 0,

then λ

lies in a proper subvariety of V and we may use induction on the

dimension. 

Notation. For B = {1, 2, . . . , n + 1} and W a vector space over k(λ) let

(W ) = Hom (

B, W ).

Corollary 4. For  large enough (depending upon V ) and λ a generic point

of V over k,



(∞)



' ker



∗

s+1,λ

, F



(`)

.



(`)



∩ δ

∗

s,λ

s−1



(∞)



(For deﬁnition of δ

∗

see [D1].)

Proof. There is a natural injection of the right hand space into the left–hand

one induced by the inclusion K

(`)

→ K

(∞)

. The left–hand space is of ﬁnite

dimension and so the mapping is surjective. 

84 BERNARD M. DWORK

We now give H

(s)

(∞)

) the structure of a diﬀerential module when

viewed as a vector space over k(λ). Let λ

, . . . , λ

be a transcendence basis

over k of k(λ). Viewing λ

t+1

, λ

t+2

etc. as dependent variables we deﬁne for

1 ≤ i ≤ t

∗

= γ

−

◦



∂

∂λ

− πx

∂h

∂λ



These operators commute with {D

∗

j,λ

}

1≤j≤n+1

and hence induce a set of

commuting operators on H

(s)

(∞)

). If λ

(1)

∈ V , then horizontal elements

are obtained by applying T

(1)

,λ

to H

(s)

(∞)

(1)

Theorem 3. Let λ

(1)

be a generic point of V . We consider all extensions to

k(λ

(1)

) of valuations of k whose restriction to k(λ

(1)

, . . . , λ

(1)

) is given by the

Gauss norm of that ﬁeld relative to λ

(1)

, . . . , λ

(1)

. For almost all such valu-

ations the horizontal elements converge for |(λ

, . . . , λ

)−(λ

(1)

, . . . , λ

(1)

)| <

Corollary 5. If k is an algebraic number ﬁeld, then H

(s)

(∞)

) is a G-

module.

Appendix A.

Let k be a ﬁeld of characteristic zero and let f (x

, . . . , x

n+1

) be a form of

degree d in n + 1 variables. If Ω is an extension of k, let us write L

Ω

for the

ring of all polynomials in x

, x

, . . . , x

n+1

of the form







+···+u

n+1



∈ Ω







We deﬁne D

= E

+ πx

, E

= x

∂

∂x

, f

= E

f. The D

are commuting

endomorphisms of L

Ω

and likewise by restricting to L

we obtain commuting

endomorphisms of that ring.

Let L

∗

Ω

(resp: L

∗

) be the adjoint space of L

Ω

(resp: L

) and K

(`)

Ω

(resp:

(`)

) the set of all ξ

∗

∈ L

∗

Ω

(resp: L

∗

) annihilated by all forms in {D

∗

, D

∗

, ...,

∗

n+1

} of degree , where D

∗

= γ

−

◦ (−E

+ πx

Again let K

(∞)

Ω

(resp: K

(∞)

) be the union

∞

[

`=1

(`)

Ω



resp :

∞

[

`=1

(`)



Finally we deﬁne H

(s)

(∞)

Ω

) (resp: H

(s)

(∞)

)) to be the s-th cohomol-

ogy group of the (cohomological) Koszul complex of D

∗

, . . . , D

∗

n+1

operating

on K

(∞)

Ω

(resp: K

(∞)

Theorem.

(i) K

(∞)

Ω

= K

(∞)

⊗ Ω

COHOMOLOGY OF SINGULAR HYPERSURFACES 85

(ii) H

(s)

(∞)

Ω

) = H

(s)

(∞)

) ⊗ Ω.

This was stated without proof as Equation (19.4) of [D1].

Proof. We ﬁrst show for  < ∞

(iii) K

(`)

Ω

= K

(`)

⊗ Ω.

We know [D1, Lemma 7.2] that dim

(`)

< ∞, dim

Ω

(`)

Ω

< ∞. We may

view each element of K

(`)

Ω

as an ∞–tuple (z

, z

, . . . ) indexed by a countable

set I. Indeed ξ

∗

∈ K

(`)

Ω

implies ξ

∗

. The sum being over all u such

that du

= u

+ · · · + u

n+1

. Here C

= π

with C

∈ Ω. Identifying the

} with the {z

}, the condition that ξ

∗

∈ K

(`)

Ω

is equivalent to an inﬁnite

set of conditions

j,i

= 0 for all j ∈ J.

Here t

j,i

∈ k, t

j,i

= 0 for almost all i, for each ﬁxed j. For ξ

∗

∈ K

(`)

we have the same set of conditions. Following a suggestion by Wan, by

elementary operations on the rows of the matrix {t

j,i

} the ﬁnite dimension

of the subspace is given by the number of zero columns in the reduced

echelon form. The echelon form is the same for the equation over Ω as over

k. I t follows that indeed dim

(`)

< ∞ ⇔ dim K

(`)

Ω

< ∞ and both are then

equal and K

(`)

Ω

= K

(`)

⊗ Ω. The ﬁrst assertion now follows.

For a vector space W we write F

(W ) = Hom (

B, W ) with B =

{1, 2, . . . , n + 1}. Then ξ

∗

∈ F

(∞)

Ω

) implies ξ

∗

a ﬁnite sum

with ξ

∗

∈ F

(∞)

) and {η

} a ﬁnite set of elements of Ω linearly indep e n-

dent over k.

If δ

∗

s+1

∗

= 0 then by linear independence δ

∗

s+1

∗

= 0 and so

ker(δ

∗

s+1

, F

(∞)

Ω

)) = ker(δ

∗

s+1

, F

(∞)

)) ⊗ Ω.

Also

∗

s−1



(∞)

Ω



= δ

∗

s−1



(∞)



⊗ Ω.

The theorem now follows from the following well known proposition. 

Proposition. Let U be a subspace of a linear k space W . Then

W ⊗ Ω/U ⊗ Ω ' (W/U) ⊗ Ω .

Appendix B. Approximation by generic points.

Lemma. Let the origin O be on an irreducible aﬃne variety V deﬁned over

a ﬁeld k of characteristic zero. Let Ω be a universal domain complete under

a rank one valuation. Then there exists a generic point of V rational over

Ω which is as close as you please to the origin.

We ﬁrst show the lemma holds if V is a curve in A

86 BERNARD M. DWORK

Proof. Let P = (x

, . . . , x

) be a generic point of V over k. Then R =

k[x

, . . . , x

] has a specialization into k given by (x

, . . . , x

) 7→ O and hence

there exists a place p of k(V ) with center O. Letting T be a uniformizing

parameter of p, each coordinate x

as element of k(V )

, the completion at

p of k(V ), is represented as a power series

= a

T + a

+ · · · + ∈ k

[[T ]]

where k

is the residue class ﬁeld at p of k(V ), a ﬁnite extension of k. This

series may have zero radius of convergence in the metric of Ω, but if we

choose (as we shall) the uniformizing parameter, T , in k(V ) then the series

represents an algebraic function of T and hence by Eisenstein’s Theorem (or

more elementarily by Clark’s Theorem) the series has a non-trivial radius of

convergence.

Since P is a generic point, these series are not all constant. We think

of P (T ) as function of T for T restricted to a small disk D(0, r

−

) in Ω–

space. Trivially P (T ) → 0 as T → 0. We may suppose x

is a non–constant

function of T . The theory of Newton polygons shows that the image of

D(0, r

−

) under x

contains elements transcendental over k. This completes

the proof for dim V = 1.

We recall [H, Chapter I, Proposition 7.1]: 

Proposition. If V is irreducible of dimension s in A

and H is a hyper-

surface not containing V then each irreducible component of H ∩ V has

dimension s − 1.

Proof of Lemma. Letting V

= V we deﬁne inductively V

⊇ V

⊇ · · · by

the condition that V

be an irreducible component of V

j−1

∩ {x | x

= 0}

which contains the origin. Since

−1 + dim V

j−1

≤ dim V

j−1

, dim V

= {0}

there exists j such that V

is a curve on V passing through the origin. 

We conclude there exists a curve V

on V passing through the origin.

Let k

⊃ k be a ﬁeld of deﬁnition of V

. By our previous treatment of

curves there exists P ∈ V

, P as close as you please to O such that k

(P )

is of transcendence degree unity over k

. Let P

be a coordinate of P of

transcendence degree unity over k

Let L = k(P

), A be the ideal of all f ∈ k[x

, . . . , x

] which are zero

everywhere on V . If g ∈ L[x

. . . , x

], g = 0 on V then g ∈ AL[x] and

hence for each automorphism τ of L/k we have g

= 0 on V . In particular

− P

cannot be zero on V as otherwise (x

− P

)

would also be zero on

V and hence P

− P

would be zero on V for every τ which is impossible as

there are nontrivial automorphisms of L/k.

Thus V does not lie in the hyperplane x

= P

and so the intersection has

an irreducible component W passing through P of dimension s−1. Let k

COHOMOLOGY OF SINGULAR HYPERSURFACES 87

a ﬁeld of deﬁnition of W , P ∈ W . By induction there exists Q ∈ W, Q − P

as small as you please with transc deg k

(Q)/k

= s − 1.

Clearly Q is as c lose as you please to O. It remains to show that s =

trans deg k(Q)/k.

Since Q

= P

, k(Q) ⊃ k(P

). Hence

s ≥ trans deg k(Q)/k = trans deg k(Q)/k(P

) + trans deg k(P

)/k

≥ trans deg k

(Q)/k

+ trans deg k

)/k

≥ (s − 1) + 1 = s,

the two inequalities being based on

if k

⊃ k then trans deg k(P

)/k ≥ trans deg k

)/k

if k

⊃ k(P

) then trans deg k(Q)/k(P

) ≥ trans deg k

(Q)/k

This completes the proof of the lemma.

Appendix C: (Generalization of Heaviside’s generalized

exponential functions).

In this article we examined the Koszul complex of {D

∗

1,λ

, . . . , D

∗

n+1,λ

} oper-

ating on K

(∞)

. In this appe ndix, we replace L

∗

0 ∗

(

n∈F



∈ π

Ω

)

and D

∗

i,λ

= γ

−

◦ (−E

+ πx

, x)) by D

0 ∗

i,λ

= −E

+ πx

(λ, x). Here

= {(u

, u

, . . . , u

n+1

)



∈ Z

n+2



= u

+ · · · + u

n+1

Thus L

0 ∗

consists of formal Laurent series in {x

} i = 1, . . . , n + 1. We

note that L

0 ∗

is adjoint to L

, the ring of Laurent polynomials with support

= u

+ · · · + u

n+1

Let D

denote the ideal of all forms of degree s in D

1,λ

, . . . , D

n+1,λ

with

coeﬃcients in k(λ). We assert that

= L + D

For s = 1 this follows by the proof of [D2, Lemma 9.7.1]. Assume the

formula valid for some given s then L

= L + D

(L + DL

) ⊂ L + D

s+1

which completes the proof by induction.

This shows that the natural m apping of L into L

induces a surjection

L/D

L → L

. We recall that K

(s)

denotes the annihilator in L

∗

L. Let K

0(s)

denote the annihilator in L

0 ∗

of D

. We now know that

the dimension of L

is ﬁnite and hence the same holds for K

0(s)

. Thus

88 BERNARD M. DWORK

by duality the mapping of K

0(s)

into K

(s)

adjoint to the natural mapping is

injective. This adjoint mapping is γ

−

Conclusion. The mapping γ

−

maps K

0(s)

into K

(s)

injectively.

We now restrict our attention to the case where h(λ, x) ∈ Q[λ, x] and

consider λ

(0)

algebraic over Q. For b > 0, c ∈ R, w a ﬁnite valuation of Ω,

let L

0 ∗

(b, c) be the set of all formal Laurent series ξ

∗

u∈F

such that

∈ Ω, and ord(B

u−v

) ≥ −b(u

+ v

) + c for all u, v, ∈ F

. Let L

0 ∗

(b) =

c∈R

0 ∗

(b, c), a Banach space. For almost all valuations of Q(λ

(0)

) we have

a completely continuous mapping of L

0 ∗

) (giving Ω a valuation extending

that of Q(λ

(0)

)) deﬁned by putting F (x) = exp π(x

h(λ

(0)

, x)−x

h(λ

(0)

, x

))

where q is the order of the residue class ﬁeld of Q(λ

(0)

) and writing

0∗

= F ◦ φ, φ : x

→ x

0∗

)

−→ L

0∗

/q)

−→ L

0∗

/q) → L

0∗

)

where b

is chosen in [0,

p−1

]. Here F means multiplication by F and the

last map is the inclusion map.

Letting L

∗

(b) = γ

−

0∗

(b) we have the completely continuous endomor-

phism of L

∗

) (for almost all valuations of Q(λ

(0)

)) given by

∗

= γ

−

◦ F ◦ φ .

By the trace formula the two mappings have the same Fredholm determi-

nant. Deﬁning

∗

[

ker



(I − zα

∗

)

, L

∗

)



0∗

[

ker



(I − zα

0∗

)

, L

0∗

)



we conclude equality of dimensions and hence

−

0∗

= W

∗

Now K

(`)

(0)

is covered by a union of spaces W

∗

and hence by γ

−

(ﬁnite

union of spaces W

0∗

) which lies in γ

−

0(`

)

(0)

for suitable 

Conclusion. γ

−

gives a bijection of K

0(∞)

(0)

onto K

(∞)

(0)

provided λ

(0)

is alge-

braic over Q.

We propose to remove the restriction that λ

(0)

be algebraic. Again let

(0)

∈ V be an algebraic point, ρ

(λ

(0)

, A) 6= 0. Excluding a ﬁnite set of

primes of Q(λ

(0)

) we choose λ generic point of V close to λ

(0)

If ξ

∗

∈ K

0(∞)

(0)

, then γ

−

∗

∈ K

(`)

(0)

for some  and hence γ

−

∗

is a ﬁnite

sum of elements of spaces W

∗

and hence is the image under γ

−

of a ﬁnite

COHOMOLOGY OF SINGULAR HYPERSURFACES 89

sum of elem ents of spaces W

0∗

. But γ

−

is injective, and hence ξ

∗

is a

sum of elements of spaces W

0∗

. Thus for almost all valuations of Q(λ

(0)

∗

∈ L

0∗

), b

p−1

. More precisely K

0(∞)

(0)

lies in L

0∗

) for all b

> 0

and almost all primes of Q. Hence for almost all primes multiplication

by exp πx



h(λ, x) − h(λ

(0)

, x)



provides an isomorphism T

(0)

,λ

of K

0(∞)

(0)

with K

0(∞)

. On the other hand, T

(0)

,λ

= γ

−

◦ T

(0)

,λ

gives an isomorphism

between K

(∞)

(0)

and K

(∞)

0(∞)

(0)

,λ

−−−−→ K

0(∞)



−



−

(∞)

(0)

−−−−→

(0)

,λ

(∞)

The horizontal arrows of this c ommutative diagram are isomorphisms.

The ﬁrst vertical arrow is also an isomorphism. It follows that the second

vertical arrow is also an isomorphism. This completes the proof.

Note. The purpose of the argument involving λ

(0)

is to show that γ

−

injective on K

0(∞)

References

[B1] C. Bertolin, G-fonctions et cohomologie des hypersurfaces singuli`eres, Bull. Aus-

tralian Math. Soc., 55 (1997), 353-383.

[B2] , G-fonctions et cohomologie des hypersurfaces singuli`eres II, Bull. Aus-

tralian Math. Soc., 58 (1998), 189-198.

[D1] B. Dwork, B. On the zeta function of a hypersurface III, Ann. Math., 83 (1966),

457-519.

[D2] , Generalized hypergeometric functions, Clarendon Press, Oxford, 1990.

[DGS] B. Dwork, G. Gerotto and F. Sullivan, An Introduction to G-functions, Annals of

Math Studies, 133 (1994), Princeton Univ. Press.

[H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York-Heidelberg-Berlin,

1977.

Received October 28, 1998.

Universita degli Studi di Pavoda

35131 Padova

Italy

E-mail address: baldassa@math.unipd.it

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

RIEMANNIAN MANIFOLDS ADMITTING ISOMETRIC

IMMERSIONS BY THEIR FIRST EIGENFUNCTIONS

Ahmad El Soufi and Sa

ıd Ilias

Given a compact manifold M, we prove that every critical

Riemannian metric g for the functi onal “ﬁrst eigenvalue of

the Laplacian” is λ

-minimal (i.e., (M, g) can be immersed

isometrically in a sphere by its ﬁrst eigenfunctions) and give

a suﬃcient condition for a λ

-minimal metric to be critical.

In the second part, we consider the case where M is the 2-

dimensional torus and prove that the ﬂat metrics correspond-

ing to square and equilateral lattices of R

are the only λ

minimal and the only critical ones.

Introduction.

Many recent works concerning the spectrum of compact Riemannian man-

ifolds have pointed out the importance of a particular class of Riemannian

metrics which we called in [5] λ

-minimal. Recall that a metric g on a

compact m-dimensional manifold M is λ

-minimal if the eigenspace E

(g)

associated to the ﬁrst nonzero eigenvalue λ

(g) of the Laplacian of g con-

tains a family f

, . . . f

of functions satisfying:

1≤i≤k

⊗ df

= g. It

follows from a well known result of Takahashi [8] that this last condition is

equivalent to the fact that the map f = (f

, . . . f

) is a minimal isometric

immersion from (M, g) into the Euclidean sphere S

k−1

of radius r =

(g)

The best known examples of λ

-minimal metrics are the standard met-

rics of rank one compact symmetric spaces (i.e., spheres and projective

spaces). More generally, any Riemannian irreducible homogeneous space is

-minimal. Also, Yau [9] conjectured that a minimal embedded hypersur-

face of a Euclidean sphere, carrying the induced metric, must be λ

-minimal.

In [2], Berger showed that the λ

-minimality of a metric g is strongly re-

lated to the extremality of g for a spectral functional involving the k-smallest

eigenvalues of the Laplacian (where k is the multiplicity of λ

(g)). Recently,

Nadirashvili [7] considered the functional λ

: g 7→ λ

(g) deﬁned on the set

of Riemannian metrics of given area on a compact surface M and showed

that the extremal metrics of this functional are λ

-minimal (here extremal-

ity is deﬁned in a generalized sense because of the non-diﬀerentiability of

92 AHMAD EL SOUFI AND SA

ID ILIAS

In the ﬁrst part of this paper we generalize Nadirashvili’s theorem to

higher dimensions (Theorem 1.1). We also give a suﬃcient condition for a

-minimal metric to be extremal for λ

(Proposition 1.1).

Using results established by us in [4] about λ

-minimal metrics we deduce

that (Corollary 1.1), if g is an extremal metric of the λ

functional then:

(i) The multiplicity of λ

(g) is at least equal to m + 1 and equality holds

only for the standard metric of Euclidean spheres.

(ii) The restriction of the λ

functional to the conformal class of g achieves

its maximum at g. In particular, the λ

functional has no local minima.

(iii) The metric g is, up to dilatation, the unique extremal metric in its

conformal class.

(iv) If g is not isometric to the standard metric of a Euclidean sphere then

any conformal diﬀeomorphism of (M, g) is an isometry.

The second part of this paper deals with the classiﬁcation of λ

-minimal

metrics and of the extremal metrics of the λ

functional. The only mani-

fold for which this classiﬁcation was available is the 2-dimensional sphere.

Indeed, on S

the standard metric is (up to dilatation) the only one to

be λ

-minimal and the only extremal metric for λ

(this follows from the

uniqueness of the conformal class on S

and property (iii) above).

The m ain theorem of Section 2 (Theorem 2.1) states that in genus one

(i.e., on the torus T

) there exists, up to dilatation, exactly two λ

-minimal

metrics: The Cliﬀord metric g

and the equilateral metric g

induced from

the Euclidean metric respec tively on R

and R

/Γ

with Γ

= Z(1, 0)⊕

Z(1/2,

√

3/2). These two metrics are also the only extremal metrics for λ

(Corollary 2.2). Moreover, we prove that for each of them, the standard

embedding (in S

for g

and S

for g

) is, up to equivalence, the only full

(minimal) isometric immersion by the ﬁrst eigenfunctions.

Note that a ﬁrst step towards this classiﬁcation was achieved by Mon-

tiel and Ros [6] who proved that the only minimal torus immersed in S

by its ﬁrst eigenfunctions is the Cliﬀord torus. They deduced that if the

aforementioned conjecture of Yau is true, then the Cliﬀord torus is the only

minimally embedded torus in S

(Lawson’s conjecture).

1. Extremal metrics for the λ

functional.

Let M b e a compact smooth manifold of dimension m ≥ 2. Denote by

(M) the set of Riemannian metrics of volume 1 on M . For any g ∈

(M), we denote by 0 < λ

(g) ≤ λ

(g) ≤ ··· ≤ λ

(g) ≤ ··· the increasing

sequence of eigenvalues of the Laplacian ∆

of g. The functional:

: R

(M) → R

g 7→ λ

(g)

ISOMETRIC IMMERSIONS BY FIRST EIGENFUNCTIONS 93

is continuous but not diﬀerentiable in general. However, for any family

)

of metrics, analytic in t, λ

) has right and left derivatives w.r.t. t.

Indeed, if (g

)

t∈]−δ,δ[

is such a family and if k is the multiplicity of λ

then there exists k analytic families Λ

1,t

, . . . , Λ

k,t

of real numbers and k

analytic families of smooth functions u

1,t

, . . . , u

k,t

such that: ∀i ≤ k and ∀t,

∆

i,t

= Λ

i,t

, Λ

i,0

= λ

) and {u

1,t

, . . . , u

k,t

} is L

)-orthonormal

(see [1] and [2] for details). Moreover, Berger [2] gave the following formula

for the derivative of Λ

i,t



t=0

= −

hq(u

), hiν

where ν

is the Riemannian volume element of g

, u

= u

i,0

, h =



t=0

h , i is the inner product induced by g

on the space S

(M) of symmetric

covariant 2-tensors of M and where for any u ∈ C

∞

(M),

q(u) = du ⊗du +

∆

From the continuity of λ

) and Λ

i,t

w.r.t. t , we have for t small enough

{Λ

i,t

}

1≤i≤k

= {λ

)}

1≤i≤k

and thus λ

) = min

1≤i≤k

{Λ

i,t

}. This proves

the left and right diﬀerentiability of λ

) and gives:

)



t=0

= min

1≤i≤k

i,t



t=0

= − max

1≤i≤k

hq(u

), hiν

and

)



t=0

−

= max

1≤i≤k

i,t



t=0

= − min

1≤i≤k

hq(u

), hiν

This suggests the following deﬁnition:

Deﬁnition 1.1. A metric g ∈ R

(M) is said to be extremal for the λ

functional if for any analytic deformation (g

)

⊂ R

(M), with g

= g, the

left and right derivatives of λ

) at t = 0 have opposite signs, i.e.,

)



t=0

≤ 0 ≤

)



t=0

−

This last condition is equivalent to :

) ≤ λ

(g) + o(t) as t → 0.

Hence our deﬁnition of extremality is a equivalent formulation of Nadi-

rashvili’s one [7].

The main result of this section is:

Theorem 1.1. If a Riemannian metric g ∈ R

(M) is extremal for λ

then

it is λ

-minimal.

94 AHMAD EL SOUFI AND SA

ID ILIAS

In the 2-dimensional cas e this result was proved by Nadirashvili [7]. Some

of the arguments in our proof are inspired by his. However, the use of the

aforementioned result of Berger makes the proof of this theorem simpler and

more transparent.

Lemma 1.1. If a metric g ∈ R

(M) is extremal for λ

then for any h ∈

(M) = {h ∈ S

(M);

hν

= 0} there exists u ∈ E

(g)\{0} such

that:

hq(u), hiν

= 0.

Proof. Suppose that g is extremal for λ

and let h ∈ S

(M). We let,

for small t, g

g+th

V (g+th)

2/m

∈ R

(M), where V (g + th) is the Riemann-

ian volume of g + th. Since

V (g + th)



t=0

hν

= 0, we ﬁnd



t=0

= h. T he extremality condition implies that the quadratic form

u ∈ E

(g) 7→

hq(u), hiν

takes on both nonpositive and nonnegative

values, and therefore it admits at least one isotropic direction. 

Proof of Theorem 1.1. Let K be the convex hull in S

(M) of {q(u), u ∈

(g)}. The set K ∪ {g} is contained in a ﬁnite dimensional subspace of

(M). We claim that g ∈ K. Indeed, if g /∈ K then, s ince K is a convex

cone, the Hahn-Banach theorem implies the existence of s ∈ S

(M) such

that:

hs, giν

> 0 and for every l ∈ K\{0},

hl, siν

< 0.

The 2-tensor ˜s = s −

(

hs,giν

)

mV (g)

g belongs to S

(M) and, for any u ∈

(g)\{0},

hq(u), ˜siν

hq(u), siν

−

mV (g)



hs, giν



|du|



< 0.

By Lemma 1.1, this contradicts the extremality of g.

Thus g ∈ K and there exists w

, ...w

∈ E

(g) such that:

g =

1≤i≤d

q(w

) =

1≤i≤d

⊗ dw





1≤i≤d

∆w





1≤i≤d



⊗ dw



(g)w

− |dw





ISOMETRIC IMMERSIONS BY FIRST EIGENFUNCTIONS 95

The traceless part of the last member of this equation must be zero. There-

fore,

1≤i≤d



⊗ dw

−

|dw



= 0,

and then:

1≤i≤d

= 1 +



m − 2



1≤i≤d

|dw

.(1)

The λ

-minimality of g will follow from the fact that

1≤i≤d

|dw

is con-

stant and equal to m. Indeed, set f =



1≤i≤d



−

(g)

. From (1) we

get:

(m − 2)∆

f = 2(m − 2)





(g)





1≤i≤d





−

1≤i≤d

|dw





= −4λ

(g)f.

This implies that f = 0 (the Laplacian being a positive ope rator). Therefore



1≤i≤d



(g)

. Replacing in (1) we obtain

1≤i≤d

|dw

= m. 

In [4] we showed that λ

-minimal metrics satisfy certain remarkable con-

formal properties. Theorem 1.1 tells us that all these properties are still

true for extremal metrics:

Corollary 1.1. Let g ∈ R

(M) be an extremal metric for λ

(i) The multiplicity of λ

(g) satisﬁes: mult(λ

(g)) ≥ m+1, where equality

holds if and only if g is isometric to a standard metric of a Euclidean

sphere.

(ii) For any g

∈ C

(g) = {g

∈ R

(M) ; g

conformal to g} we have

) ≤ λ

(g), and equality holds if and only if g

is isometric to g.

In pa rticular, the functional λ

does not admit a local minimum in

(M).

(iii) The metric g is, up to isometry, the only extremal metric of λ

(g).

(iv) If (M, g) is not isometric to a Euclidean sphere then any conformal

diﬀeomorphism of (M, g) is an isometry.

The following is a converse to Theorem 1.1.

Proposition 1.1. Let g ∈ R

(M) and assume there exists an L

(g)-ortho-

normal basis {φ

, . . . , φ

} of E

(g) such that the 2-tensor

1≤i≤k

dφ

⊗dφ

is proportional to g. Then g is extremal for λ

96 AHMAD EL SOUFI AND SA

ID ILIAS

Proof. Let (g

)

⊂ R

(M) be a family of metrics analytic in t with g

= g

and set h =



t=0

. With the same notation as above we have for small t:

1≤i≤k

) =

1≤i≤k

i,t

Therefore,

1≤i≤k

) is diﬀerentiable at t = 0 and

1≤i≤k

)



t=0

1≤i≤k

i,t



t=0

= trace Q

where Q

is the quadratic form deﬁned on E

(g) by:

(u) =

hq(u), hiν

and where the trace of Q

is taken w.r.t the L

inner product induced by g.

Now

trace Q

1≤i≤k

(φ

)

1≤i≤k

dφ

⊗ dφ

, h

1≤i≤k



∆φ

, h



Since

1≤i≤k

dφ

⊗dφ

is proportional to g and

hg, hiν

V (g

)



t=0

0 we have

1≤i≤k

dφ

⊗ dφ

, h

= 0. Moreover, by Takahashi’s theo-

rem

1≤i≤k

is constant. Therefore, trace Q

=0 and

1≤i≤k

)



t=0

= 0. The extremality of g then follows from the inequality λ

) ≤

1≤i≤k

) which is an equality at t = 0. 

Remarks.

1) It is known that compact irreducible homogeneous Riemannian spaces

satisfy the hypothesis of Proposition 1.1 (see [8]). Thus, their standard

metrics are extremal for λ

2) We restricted ourselves to λ

. Nevertheless, the results of this para-

graph can be carried over to the case of higher eigenvalues.

2. λ

-minimal and extremal metrics on the torus.

Let (M, g) be an orientable compact surface of genus one endowed with a

Riemannian metric g. It is well known that there exists a lattice Γ of R

such that (M, g) is conformally equivalent to the torus



/Γ, g



, where

is the ﬂat me tric induced from the Euclidean metric on R

. The Cliﬀord

torus



= R

/Γ

, g

= g



with Γ

= Z(1, 0) ⊕ Z(0, 1), and the equi-

lateral torus



= R

/Γ

, g

= g



with Γ

= Z(1, 0) ⊕ Z(1/2,

√

3/2),

ISOMETRIC IMMERSIONS BY FIRST EIGENFUNCTIONS 97

each admit a natural homothetic minimal embedding into a sphere. Thes e

embeddings, denoted by φ

and φ

, are those induced on T

and T

from

: R

→ S

, where

(x, y) =

√

(exp 2iπx, exp 2iπy), and

: R

→ S

where

(x, y)=

√



exp 4iπy/

√

3, exp 2iπ(x − y/

√

3), exp 2iπ(x + y/

√



Theorem 2.1. Let (M, g) be a compact orientable surface of genus one and

suppose that there exists a full isometric immersion φ = (φ

, . . . , φ

n+1

) from

(M, g) in the n-dimensional unit sphere S

such that ∀i ≤ n+1, φ

∈ E

(g).

Then either:

(i) (M, g) is isometric to the normalized Cliﬀord torus (T

, 2π

), n = 3

and φ is equivalent to φ

, or

(ii) (M, g) is isometric to the normalized equilateral torus (T

8π

n = 5 and φ is equivalent to φ

Recall that an immersion φ into S

is full if its image is not contained in a

great sphere of S

. Two immersions φ and ψ into S

are called equivalent if

there exists an isometry R of S

such that φ = R ◦ψ. A direct consequence

of Theorem 2.1 is:

Corollary 2.1. A compact genus one orientable surface (M, g) is λ

-mini-

mal if and only if it is homothetic to (T

, g

) or (T

, g

As the metrics g

and g

trivially satisfy the hypothesis of Proposition

1.1 we have the following:

Corollary 2.2. Let M be a compact orientable surface of genus one. A

metric g on M is extremal for λ

if and only if (M, g) is homothetic to

, g

) or (T

, g

The proof of Theorem 2.1 is base d on the following Propositions 2.1 and

2.2 which are valid in a more general setting.

Proposition 2.1. Let (M, g) be a n-dimensional compact Riemannian ho-

mogeneous manifold non homothetic to S

. If a metric g = fg

, conformal

to g

, is λ

-minimal, then f is constant on M.

Proof. As (M, g) is λ

-minimal non homothetic to S

then any conformal

diﬀeomorphism of (M, g) is an isometry (cf. [4]). It follows that any isome-

try of (M, g

) is also an isometry of (M, g). Thus the function f is invariant

under the isometry group of (M, g

). The result follows from the homogene-

ity of (M, g

). 

Proposition 2.2. Let η

, η

, . . . , η

be N continuous functions on a do-

main Ω of R

and assume that the N

functions: 2η

(1 ≤ j ≤ N), η

+ η

and η

−η

(1 ≤ k < l ≤ N) are non-constant and mutually distinct modulo

2π. If φ = (φ

, . . . , φ

n+1

) is a map from Ω to S

such that all its compo-

nents φ

are in the vector space generated by {cos η

, sin η

, 1 ≤ j ≤ N},

98 AHMAD EL SOUFI AND SA

ID ILIAS

then there exists an isometry R of S

such that

R ◦ φ = (α

exp iη

, α

exp iη

, . . . , α

exp iη

, 0, . . . , 0) ,

where r ≤ (n + 1)/2, j

, . . . , j

∈ {1, . . . , N} and α

, . . . , α

are positive

constants satisfying

1≤j≤r

= 1. In particular, R(φ(Ω)) ⊂ S

(α

) ×

(α

) × ··· × S

(α

) × {0}.

The proof of this proposition is quite elementary and can be omitted.

Proof of Theorem 2.1. In view of Proposition 2.1, it suﬃces to consider the

case where the metric g is ﬂat. It is well known that there exists (a, b) ∈ R

;

0 ≤ a ≤

, b > 0 and a

+ b

≥ 1, such that (M, g) is homothetic to



a,b

= R

/Γ(a, b), g

= g

Γ(a,b)



with Γ(a, b) = Z(1, 0) ⊕ Z(a, b) (cf. [3]).

Now the existence of an isometric immersion from (M, g) into the unit sphere

by the ﬁrst eigenfunctions implies that λ

(g) = 2. Since λ

) =

4π

(M, g) is in fact isometric to



a,b

2π



. Let E

a,b

be the ﬁrst eigenspace

of g

and φ :



a,b

2π



→ S

a full isometric immersion whose compo-

nents φ

∈ E

a,b

• If a

+ b

> 1 then the dimension of E

a,b

is 2 and there is no such φ.

• If a

+ b

= 1 and (a, b) 6= (1/2,

√

3/2) then E

a,b

is generated by cos η

sin η

, j ≤ 2, with η

(x, y) =

2πy

and η

(x, y) = 2π



x −



. From

Proposition 2.2, it follows that n = 3 and, up to an isometry of S

φ has the form φ = (α

exp(iη

), α

exp(iη

)) with α

> 0, α

> 0

and α

+ α

= 1. As φ is isometric we deduce that a = 0, b = 1 and

= α

√

2/2. Thus (M, g) is isometric to (T

, 2π

) and φ is

equivalent to φ

• If (a, b) = (1/2,

√

3/2) then E

a,b

is generated by cos η

, sin η

, j ≤ 3,

where η

(x, y) = 4πy/

√

3, η

(x, y) = 2π



x −

√



and η

(x, y) =

2π



x +

√



. As before n ≤ 5 and, up to isometry, φ = (α

exp(iη

), α

exp(iη

)) where α

, α

and α

are nonnegative con-

stants such that α

+ α

= 1. As φ is isometric we obtain

= α

√

3/3. Thus φ is equivalent to φ



References

[1] S. Bando and H. Urakawa, Generic properties of the eigenvalue of Laplacian for com-

pact Riemannian manifolds, Tˆohoku Math. J., 35 (1983), 155-172.

[2] M. Berger, Sur les premi`eres valeurs propres des vari´et´es riemanniennes, Compositio.

Math., 26 (1973), 129-149.

ISOMETRIC IMMERSIONS BY FIRST EIGENFUNCTIONS 99

[3] M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une vari´et´e riemannienne, Lec-

ture Notes in Math., Vol. 194, Springer, 1971.

[4] A. El Souﬁ and S. Ilias, Immersions minimales, premi`ere valeur propre du laplacien et

volume conforme, Math. Ann., 275 (1986), 257-267.

[5] , Majoration de la seconde valeur propre d’un op´erateur de Schr

ødinger sur une

vari´et´e compacte et applications, J. of Funct. Anal., 103(2) (1992), 294-316.

[6] S. Montiel and A. Ros, Minimal immersions of surfaces by the ﬁrst eigenfunctions and

conformal area, Invent. Math., 83 (1986), 153-166.

[7] N. Nadirashvili, Berger’s isoperimetric problem and minimal immersions of surfaces,

Geom. and Funct. Anal., 6 (1996), 877-897.

[8] T. Takahashi, Minimal immersions of riemannian manifolds, J. Math. Soc. Jap., 18

(1966), 380-385.

[9] S.-T. Yau, Problems section. Seminar on Diﬀerential Geometry, Ann. Math. Stud.,

Vol. 102, Princeton University Press, 1982.

Received October 20, 1998 and revised June 15, 1999.

Laboratoire de mathematiques et physique theo rique

Universite de Tours

Parc de Grandmont

37200 Tours

France

E-mail address: elsouﬁ@univ-tours.fr

Laboratoire de mathematiques et physique theo rique

Universite de Tours

Parc de Grandmont

37200 Tours

France

E-mail address: ilias@univ-tours.fr

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

THE RUBINSTEIN–SCHARLEMANN GRAPHIC OF A

3-MANIFOLD AS THE DISCRIMINANT SET OF A

STABLE MAP

Tsuyoshi Kobayashi and Osamu Saeki

We show that Rubinstein–Scharlemann graphics for 3-mani-

folds can be regarded as the images of the singular sets

(: discriminant set) of stable maps from the 3-manifolds into

the plane. As applications of our understanding of the graphic,

we give a method for describing Heegaard surfaces in 3-mani-

folds by using arcs in the plane, and give an orbifold version

of Rubinstein–Scharlemann’s setting. Then by using this set-

ting, we show that every genus one 1-bridge position of a non-

trivial two bridge knot is obtained from a 2-bridge position in

a standard manner.

1. Introduction.

In this paper, we show that Rubinstein-Scharlemann graphics for 3-manifolds

can be regarded as the images of the singular sets (: discriminant set) of sta-

ble maps from the 3-manifolds into the plane, and as applications, we give

a method for describing Heegaard surfaces in 3-manifolds by using arcs in

the plane, and give an orbifold version of Rubinstein-Scharlemann’s setting.

Then by using this setting, we show that every genus one 1-bridge position

of a non-trivial two bridge knot is obtained from a 2-bridge position in a

standard manner.

In [18], Rubinstein-Scharlemann introduced a powerful machinery, which

is called a graphic, for studying Heegaard s plittings of 3-manifolds, and suc-

ceeded to obtain deep results on the Reidemeister-Singer distance of two

strongly irreducible Heegaard splittings of a 3-manifold. We note that Ru-

binstein and Scharlemann derived a graphic from two Heegaard splittings of

a 3-manifold via Cerf theory [5]. Then the purpose of this paper is to intro-

duce another way for understanding the graphic. That is, we show that we

can regard a graphic as the image of the singular set of a “stable map”(for

deﬁnition, see Sect. 3) from the 3-manifold into the plane R

(Theorem 4.2).

An immediate consequence of this is that we can regard a Heegaard sur-

face as the preimage of an arc in R

, and as an application of our understand-

ing, we ﬁrst give a method for instructing a procedure for deforming one

Heegaard surface to the other by using the arcs as above (Proposition 5.4),

101

102 T. KOBAYASHI AND O. SAEKI

and describe how stabilization works in this setting (Proposition 5.6). In

[19], Rubinstein and Scharlemann give a generalization of results in [18]

for 3-manifolds with boundary. As the second application, we will give an-

other formulation for generalizing the idea in [18] for link spaces. In fact,

we will introduce an orbifold version of the Rubinstein-Scharlemann type

argument(Sect. 6), and, by using this, we show that any genus one 1-bridge

position of a 2-bridge knot is obtained from a 2-bridge position in a standard

manner (Theorem 8.2).

2. Rubinstein-Scharlemann graphic.

Throughout this paper, we work in the diﬀerential category, and for standard

terminology in 3-dimensional topology, we refer to [9], and [11].

In this section, we quickly review the setting of Rubinstein-Scharlemann’s

paper [18].

Let M be a closed orientable 3-manifold.

Deﬁnition 2.1. We say that a decomposition M = A ∪

B is a (genus g)

Heegaard splitting of M if A, B are 3-dimensional genus g handlebodies in

M such that M = A ∪ B, A ∩ B = ∂A = ∂B = P . Then P is called a (genus

g) Heegaard surface of M.

Deﬁnition 2.2. A disk D properly embedded in a handlebody H is called

a meridian disk of H if ∂D is an essential simple close d curve in ∂H.

Deﬁnition 2.3. A Heegaard splitting M = A ∪

B is stabilized, if there

are meridian disks D

, D

of A, B respectively such that ∂D

and ∂D

intersects transversely in a single point.

Remark 2.4. We note that a genus g Heegaard splitting M = A ∪

B is

stabilized if and only if there exists a genus g −1 Heegaard splitting A

∪

such that A ∪

B is obtained from A

∪

by adding a “trivial” handle.

Then we say that M = A ∪

B is obtained from A

∪

by a stabilization.

Deﬁnition 2.5. A Heegaard splitting M = A ∪

B is reducible, if there

exist meridian disks D

, D

of A, B respectively such that ∂D

= ∂D

Deﬁnition 2.6. A Heegaard splitting M = A ∪

B is weakly reducible, if

there exist meridian disks D

, D

of A, B res pectively such that ∂D

∩

∂D

= ∅.

Remark 2.7. It is easy to see that if a Heegaard splitting M = A ∪

is reducible then it is weakly reducible. And it is also easy to see that if

M = A ∪

B is stabilized and is not a genus one Heegaard splitting of the

3-sphere S

, then it is reducible.

Remark 2.8. It is known, by Haken [8], that if M is reducible (that is,

if M is a connected sum of two 3-manifolds which are not S

), then any

Heegaard splitting of M is reducible.

RUBINSTEIN–SCHARLEMANN GRAPHIC 103

Remark 2.9. It is known, by Casson-Gordon [4], that if a Heegaard split-

ting M = A ∪

B is weakly reducible, then either it is reducible, or M

contains an incompressible surface.

Setting of Rubinstein-Scharlemann graphic.

Let A∪

B, X ∪

Y be a pair of Heegaard splittings of a closed 3- manifold

M. Let Θ

, Θ

be spines of A, B, X, Y respectively such that

(except for genus 0, or 1 Heegaard splittings) each vertex of Θ

, Θ

has valency 3 (see Figure 2.1). Note that for a genus 0 handleb ody (:the

3-ball) B

, we let the spine of B

be a point in IntB

, and for a genus 1

handlebody (:solid torus), we let the spine be a core circle of the solid torus.

Then M −(Θ

∪Θ

) is homeomorphic to P ×(0, 1), where P ×{ε} is close to

, and P ×{1−ε} is close to Θ

for a small ε > 0. We let P

be the s urface

in M corresponding to P ×{s}. Then, by regarding P

= Θ

, and P

= Θ

we obtain a continuous map H : P × I → M such that H(P, s) = P

, and we

call this a sweep-out associated to A ∪

B. Similarly we obtain a sweep-out

G : Q × I → M associated to X ∪

Y , and set G(Q, t) = Q

Here we may suppose that Θ

∪ Θ

and Θ

∪ Θ

, Θ

∪ Θ

and G, and

∪ Θ

and H are in general positions. This implies that “H(P × [0, ε]) ∪

H(P × [1 − ε, 1]) and G”, “G(Q × [0, ε]) ∪ G(Q × [1 − ε, 1]) and H” are in a

“standard” position. That is:

Regard G(Q×(0, 1))

−1

→ Q×(0, 1)

proj.

→ (0, 1) as a height function.

Then except for a neighborhood of the maxima and minima (with

respect the height function) and vertices of Θ

, each component

of H(P × [0, ε]) ∩ Q

is a meridian disk intersecting Θ

in one

point (for a small ε > 0), and in the neighborhoods Q

looks as

in Figure 2.2. The same picture holds for the other pair.

Figure 2.1.

104 T. KOBAYASHI AND O. SAEKI

Figure 2.2.

Then, by Cerf [5], we see that for “generic” sweep-outs H, G we obtain

a stratiﬁcation of Int(I × I) which consists of four parts below.

Regions: Each region is a component of the subset of Int(I × I) consist-

ing of values (s, t) such that P

and Q

intersect transversely, and this

is an open set.

Edges: Each edge is a component of the subset consisting of values

(s, t) such that P

and Q

intersect transverse ly except for one non-

degenerate tangent point. The tangent point is either a “center” or

a “saddle”. Edge is a 1-dimensional s ubset of Int(I × I), which is

monotonously increasing or decreasing.

Figure 2.3.

Figure 2.4.

Crossing vertices: Each crossing vertex is a component of the subset

consisting of points (s, t) such that P

and Q

intersect transversely

except for two non-degenerate tangent points. Crossing vertex is an

isolated point in Int(I × I). In a neighborhood of a crossing vertex,

four edges are coming in, where one can regard the crossing vertex as

RUBINSTEIN–SCHARLEMANN GRAPHIC 105

the intersection of two edges `

, `

with the signs of the slopes of `

and `

are either diﬀerent or the same.

Figure 2.5.

Figure 2.6.

Birth-death vertices: Each birth-death vertex is a component of the

subset consisting of points (s, t) such that P

and Q

intersect trans-

versely except for a single degenerate tangent point. In particular,

there is a parametrization (λ, µ) of I × I such that P

= {(x, y, z)|z =

0}, and Q

= {(x, y, z)|z = x

+ λ + µy + y

}. A birth-death vertex is

an isolated point in Int(I × I), and in a neighborhood of a birth-death

vertex, two edges `

, `

are coming in, with one from c enter tangency,

the other from saddle tangency, and the signs of the slopes of `

and

the same.

106 T. KOBAYASHI AND O. SAEKI

Figure 2.7.

Let Γ be the union of edges and vertices above. By the above, Γ is a

1-complex in Int(I × I). Since we have assumed that H, G are standard in

a regular neighborhood of Θ

∪ Θ

, we see that the 1-complex

Γ naturally extends to ∂(I × I). We abuse Γ to denote this 1-complex, and

we call Γ a graphic (obtained from the sweep-outs H, G).

Figure 2.8

Example 2.10. We show that there exist inﬁnitely many 3-manifolds, and

a pair of Heegaard splittings, say A ∪

B, X ∪

Y , of each 3-manifold such

that the corresponding graphic is as in Figure 2.9.

Figure 2.9.

Note that the picture admits 4-fold (Z

⊕ Z

) symmetry, and we will give

an explicit description of the Heegaard surfaces belonging to the lower-left

RUBINSTEIN–SCHARLEMANN GRAPHIC 107

quarter of I ×I, which can be naturally extended to the whole picture under

the above symmetry.

We may regard the Heegaard surface P as P

1/2

, and Q as Q

1/2

. Then

A and A ∩ Q

(0 ≤ ε ≤ 1/2) look like as follows. (Here we suppose that

A admits a symmetry generated by ϕ

, ϕ

in Figure 2.10, where ϕ

is an

orientation preserving involution which changes the right side and the left

side of A with the ﬁxed point set an arc properly embedded in A, and ϕ

an orientation reversing involution which changes the right side and the left

side of A with the ﬁxed point set a disk properly embedded in A.)

A ∩ Q

(= A ∩ Θ

) is a 1-complex as in Figure 2.10. Then, when ε is

suﬃciently small, A ∩ Q

is the frontier of a regular neighborhood of the 1-

complex (the surface is homeomorphic to a torus with one hole). If we make

ε bigger, then we come to the point (1) of Figure 2.9 and simlutaneously

four points in the boundary of a torus Q

tend to four directions as in (1)

of Figure 2.10.

Figure 2.10.

Then, by m aking ε bigger further, we come to the point (2) and, then, (3)

(: ε = 1/2), where the corresponding ﬁgures of A∩Q

look as in Figure 2.11.

When we come to the point (2), the boundary component touches itself

simultaneously in two places. In the right side, a band is produced, and, in

the left side, the surface is boundary compressed when we pass the point (2).

Note that A ∩ Q

1/2

is a vertical surface (which is a disk with two holes),

which is located in the middle of A, that is, A ∩ Q

1/2

is invariant under ϕ

and ϕ

exchanges the components of A − Q

1/2

Figure 2.11.

108 T. KOBAYASHI AND O. SAEKI

Figures 2.12 describes the deformations (4) → (5). When we come to

(5) from (1) with passing the edge containing (4), a torus with one hole is

boundary compressed and becomes an annulus.

Figure 2.12.

Figure 2.13 describes the surface of (6). When we come to (6) from (5),

two points of a boundary component of an annulus tend to the right side

and touch and a band is produced, and simultaneously two points contained

in diferent boundary components of an annulus tend to the back of A and

the surface is boundary copressed.

Figure 2.13.

Figures 2.14 and 2.15 des cribe the surfaces of (7), (8), and (9). At (7)

Q ∩ A is an inessential disk properly embedded in A. When we come to (8)

from (7), two points of the boundary of a disk tend to the right side and

touch at a middle part of A, and simultaneously two points in the boundary

of a disk tend to the back of A and the surface is boundary comressed. As

a result, an inessential disk be come s a separating essential disk in A.

Figure 2.14.

RUBINSTEIN–SCHARLEMANN GRAPHIC 109

Figure 2.15.

Figure 2.16 describes (10). When we come to (10) from (7) with passing

an edge, an inessential disk is boundary compressed and becom es two non-

separating disks.

Figure 2.16.

Let A ∩ Q

1−ε

= ϕ

(A ∩ Q

) (0 ≤ ε ≤ 1/2), which gives the whole descrip-

tion of Q

(0 ≤ t ≤ 1) in A.

Let B be a copy of A, and f : A → B the homeomorphism induced by the

identiﬁcation, and φ = f|

∂A

: ∂A → ∂B the corresponding homeomorphism.

Let F be the pattern on ∂A induced by Q

’s. Le t M be the 3-manifold

obtained by attaching A to B by the homeomorphism φ ◦ ϕ

, where ϕ

∂A → ∂A is isotopic to ϕ

with ϕ

(F) = F, and ϕ

) = `

, ϕ

) = `

(Note that M is actually the connected sum of two S

× S

’s.) We note

that F is invariant under ϕ

, and, hence, the surfaces Q

(0 < t < 1) in A

are matched to the surfaces f ◦ ϕ

) (0 < t < 1) in B to make a system

of closed surfaces, say Q

again, in M. It is directly observed from the

pictures that Q

gives a sweep out G : Q × I → M, and, by construction,

we immediately see that the corresponding graphic is as in Figure 2.9.

Then let `

, `

be the components of ∂(A ∩ Q

1/2

) as in Figure 2.11, and

: ∂A → ∂A (i = 1, 2) the Dehn twist along `

. For a pair of integers

(p, q), we let M

(p,q)

be the manifold obtained by attaching B to A by the

homeomorphism φ ◦ D

◦ D

◦ ϕ

: ∂A → ∂B. Since there is a regular

neighborhood N(`

, ∂A) such that F restricted to N(`

, ∂A) is a foliation

by circles parallel to `

, we can arrange so that the conﬁguration of the

sweep-outs H and G are respected in M

(p,q)

(and, hence, the corresponding

graphic is the same as above). It is easy to see that M

(p,q)

is a connected

110 T. KOBAYASHI AND O. SAEKI

sum of two lens spaces L(p, 1), and L(q, 1), which implies the existence of

inﬁnitely many examples.

We note that it is easy to see from Figure 2.11, that the Heegaard surfaces

are isotopic in each of the examples.

Example 2.11. By using the arguments in Example 2.10, we show that

there exist inﬁnitely many 3-manifolds, and a pair of Heegaard splittings,

say A∪

B, X ∪

Y , of each 3-manifold such that the corresponding graphic

is as in Figure 2.17.

Figure 2.17.

As in Figure 2.10, Q

is a torus with one hole which is the frontier of a

regular neighborhood of a 1-complex for a small ε > 0. Four points in the

boundary of the torus with one hole tend to four directions according as the

expansion of the regular neighborhood.

Figure 2.18.

When we come to (3a) from (1) with passing (2), a band which goes

around the right handle twice is produced, and a slit in the surface which

goes around the left handle twice occurs to produce a boundary compression.

As a result, a torus with one hole becomes a disk with two holes.

RUBINSTEIN–SCHARLEMANN GRAPHIC 111

Figure 2.19.

Figure 2.20a.

Note that in (3b), Q

1/2

is invariant with respect to the involution ϕ

Figure 2.20b.

When we come to (6) from (3b) with passing (4), two boundary compres-

sions occur, and a disk w ith two holes becomes a separating essential disk

in A.

Figure 2.21.

112 T. KOBAYASHI AND O. SAEKI

In the following, we show pictures with turning back to front for the

convenience of drawing.

When we come to (5) from (3b) with passing an edge, the right band is

boundary compressed and a disk with one hole becomes an annulus.

When we come to (6) from (5) with passing an edge, an annulus is bound-

ary compressed to become a separating essential disk.

Figure 2.22.

Figure 2.23.

When we come to (6) from (1) with passing (7), two bands, one of which

goes around the left handle once and the other goes around the left handle

twice are produced.

Figure 2.24.

When we come to (8) from (1) with passing an edge, a torus with one

hole is boundary compressed and the punctured torus become an annulus.

When we come to (8) from (6) with passing an edge, a band which goes

around the left handle once is attached and the disk becomes an annulus.

RUBINSTEIN–SCHARLEMANN GRAPHIC 113

Figure 2.25.

When we come to (6) from (10) with passing (9), two points of the bound-

ary of an inessential disk tend to the right side and touch at a middle part

of A, and simultaneously two points in the boundary of a disk tend to the

back of A and touch to produce a boundary compression. As a result, an

inessential disk becomes a separating essential disk in A.

Figure 2.26.

When we come to (11) from (10) with passing an edge, an inessential disk

is boundary compressed and becomes two non-separating disks.

Figure 2.27.

Note that, by using the arguments in the previous example, we can show

that such 3-manifolds are obtained from A and B by pasting their boundaries

applying the Dehn twists along ` in Figure 2.20b. Note that, as a result of

this construction, we obtain 3-manifolds each of which is a union of two

Seifert ﬁbered spaces with orbit space a disk with two exceptional ﬁbers

of index two, and the exterior of a (2, 2n)-torus link (see [13]). Note that

except in one case (case n = 0) they are Haken manifolds.

114 T. KOBAYASHI AND O. SAEKI

3. Stable maps.

The purpose of this section is to show that any diﬀerentiable map from an n-

manifold into a surface can be deformed to an “excellent” (:stable) map, and

this assertion is an es sential part of this paper. In the following, manifolds

have countable basis and all manifolds and maps are assumed to be C

∞

Let M be a connected n-dimensional manifold (possibly with boundary)

with n ≥ 2 and N a surface without boundary. For a smooth map f : M →

N, S(f) denotes the singular set of f; i.e., S(f) is the set of the points in M

where the rank of the diﬀerential df is strictly less than 2. The discriminant

set is the image of the singular set, f(S(f )). We denote by C

∞

(M, N) the

space of the smooth maps of M into N endowed with the Whitney C

∞

topology (ﬁne topology) (see [7, 10]).

Deﬁnition 3.1. A smooth map f : M → N is stable if there exists a

neighborhood U of f in C

∞

(M, N) such that for each g ∈ U there exist

diﬀeomorphisms H : M → M and h : N → N which make the following

diagram commutative.

−−−→ N



M −−−→

Deﬁnition 3.2. Let f : M → N be a proper smooth map of an n-dimensio-

nal manifold M (n ≥ 2) into a surface N without boundary. For an open

set U of N, we say that f is excellent on U, if f

−1

(U) ∩ ∂M = ∅ and

the following conditions are s atisﬁed: For all p ∈ f

−1

(U), there exist local

coordinates (u, x, y

, · · · , y

n−2

) centered at p and (X, Y ) centered at f(p)

such that f has one of the following forms:

) X ◦ f = u, Y ◦ f = x (p : regular point)

) X ◦ f = u, Y ◦ f = ±x

n−2

i=1

±y

(p : fold point)

) X ◦ f = u, Y ◦ f = ux − x

n−2

i=1

±y

(p : cusp point);

and

) If p ∈ f

−1

(U) is a cusp point, then f

−1

(f(p)) ∩ S(f) = {p},

) f|

(S(f )∩f

−1

(U)−{cusp points})

is an immersion with normal crossings.

RUBINSTEIN–SCHARLEMANN GRAPHIC 115

Note that it is well known that a prop e r smooth map f : M → N of a

manifold M with ∂M = ∅ is stable if and only if f is excellent on N (see

[7, 14]). The terminology “excellent”comes from [22, §2].

The main purpose of this section is to prove the following.

Theorem 3.3. Let f : M → N be a proper smooth map. Suppose that F

is a closed 2-dimensional submanifold (possibly with boundary) of N such

that f

−1

(Int F ) ⊃ ∂M. Furthermore we suppose that f is excellent on a

neighborhood of ∂F and that f is transverse to ∂F . Furthermore, let V

be an arbitrary open neighborhood of f in C

∞

(M, N). Then there exists a

smooth map g : M → N such that

(1) g ∈ V,

(2) g = f on f

−1

(F ) = g

−1

(F ),

(3) g is excellent on a neighborhood of N − Int F .

Remark 3.4. In Theorem 3.3 the condition (2) is essential. In fact, if we

drop the condition (2), it has already been well known.

In the following, we use the following notation. For manifolds X and Y ,

(X, Y ) denotes the k-jet bundle over X×Y , i.e., J

(X, Y ) = {(x, y, j

(x))|

x ∈ X, y ∈ Y ∃f : X → Y : C

∞

s.t. f(x) = y}. For integers s ≥ 1 and k ≥ 0,

(X, Y ) denotes the s-fold k-jet bundle ([7, p. 57]). We denote by X

(s)

the subset of X

= X × · · · × X (the s-fold product space of X) consisting

of the elements (x

, · · · , x

) such that x

6= x

for i 6= j. We denote by π :

(X, Y ) → X

(s)

× Y

the canonical projection and by π

: J

(X, Y ) → Y

the natural projection to the target. We se t ∆

= {(y, · · · , y) ∈ Y

} and

d : ∆

→ Y is the natural identiﬁcation map. Furthermore, for a smooth

map f : X → Y , j

f : X

(s)

→ J

(X, Y ) denotes the s-fold k-jet of f . (For

details, see [7, Chapter II, §4].)

In order to prove Theorem 3.3, we need the following.

Proposition 3.5. Let f : X → Y be a smooth map between manifolds

(Y need not be a surface). Let W be a submanifold of J

(X, Y ) such that

(W ) ⊂ ∆

. Suppose that U is an open subset of Y and that V is an open

neighborhood of f in C

∞

(X, Y ). Then there exists a smooth map g : X → Y

such that

(1) g ∈ V,

(2) g = f on f

−1

(U) = g

−1

(U),

(3) j

g is transverse to W on W ∩ π

−1

(Y − U)).

Proof. Set W

= W ∩ π

−1

(Y − U)), which is an open submanifold of

W . Then there exists a countable family {W

}

∞

r=1

of open sets of W

with

the following properties (a)-(f).

(a) ∪

∞

r=1

= W

116 T. KOBAYASHI AND O. SAEKI

(b) W

⊂ W

, where W

denotes the closure of W

in J

(X, Y ).

is compact.

(d) There exist coordinate neighborhoods U

r,1

, · · · , U

r,s

in X and V

r,1

, . . . ,

r,s

in Y such that {U

r,i

}

i=1

are mutually disjoint and π(W

) ⊂ U

r,1

× · · · ×

r,s

× V

r,1

× · · · × V

r,s

(e) U

r,i

is compact for 1 ≤

∀

i ≤ s.

(f) V

r,i

∩ U = ∅ for 1 ≤

∀

i ≤ s.

Using this family {W

}

∞

r=1

in the argument of [7, pro of of Theorem 4.13

(p. 58)] or [15, pp. 311-312], we see that there exists a smooth map g

X → Y such that j

: X

(s)

→ J

(X, Y ) is transverse to W on W

and that

= f on f

−1

(U) = g

−1

(U) in an arbitrary neighborhood of f in C

∞

(X, Y ).

Thus, putting

f,U

= {g ∈ C

∞

(X, Y ) : g = f on f

−1

(U) = g

−1

(U)},

we see that

= C

f,U

∩ {g ∈ C

∞

(X, Y ) : j

g is transverse to W on W

}

is dense in C

f,U

. On the other hand, D

is open by [7, Lemma 4.14 (p. 57)].

The proposition is proved if we show that ∩

∞

r=1

is dense in C

f,U

. Thus we

have only to show that C

f,U

is a Baire space (see [7, Deﬁnition 3.2 (p. 44)]).

First note that C

f,U

is a closed subset of C

∞

(X, Y ). Then by imitating the

proof of [7, Proposition 3.3 (p. 44)], we see easily that C

f,U

is a Baire space.

This completes the proof. 

Let M and N be as in Theore m 3.3. We consider some submanifolds of the

(multi-)jet bundles as follows. For the jet bundle J

(M, N), we consider the

four submanifolds Σ

n−1,0

, Σ

n,0

, Σ

n−1,1,0

and Σ

n−1,1,1,0

as deﬁned in [1] (or [7,

p. 156, Sect. 5]. Note that their codimensions are equal to n − 1, 2n, n and

n+1 respectively by [1, Theorem (6.2)]. For the multi-jet bundle J

(M, N),

we consider

= {(j

f(p), j

g(q)) : f(p) = g(q), j

f(p) ∈ Σ

n−1,0

, j

g(q) ∈ Σ

n−1,0

= {(j

f(p), j

g(q)) : f(p) = g(q), j

f(p) ∈ Σ

n−1,0

, j

g(q) ∈ Σ

n−1,1,0

= {(j

f(p), j

g(q)) : f(p) = g(q), j

f(p) ∈ Σ

n−1,1,0

, j

g(q) ∈ Σ

n−1,1,0

For the multi-jet bundle J

(M, N), we consider

= {(j

f(p), j

g(q), j

h(r)) : f(p) = g(q) = h(r),

f(p), j

g(q), j

h(r) ∈ Σ

n−1,0

Note that S

, S

and S

are easily seen to be submanifolds and that

their codimensions are equal to 2n, 2n + 1, 2n + 2 and 3n + 1 respectively.

For a smooth map f : M → N, we have the following facts:

(1) j

f is transverse to Σ

n−1,0

, Σ

n,0

, Σ

n−1,1,0

and Σ

n−1,1,1,0

if and only if

f exhibits only fold and cusp points as its singularities.

RUBINSTEIN–SCHARLEMANN GRAPHIC 117

(2) Suppose f satisﬁes (1). Then j

f is transverse to S

if and only if

(S(f )−{cusp points})

has no multiple points of multiplicity greater than two.

(3) Suppose f satisﬁes (1) and (2). Then j

f is transverse to S

if and

only if f|

(S(f )−{cusp points})

is an immersion with normal crossings (see [7,

Proposition 5.6 (p. 158)]).

(4) Suppose f satisﬁes (1). Then j

f is transverse to S

and S

if and

only if for every cusp point p of f, we have f

−1

(f(p)) ∩ S(f) = {p}.

Using the above facts, we obtain the following.

Lemma 3.6. Let f : M → N be a proper smooth map of an n-dimensional

manifold M (n ≥ 2) into a surface N. For an open set U of N, f is excellent

on U if and only if f

−1

(U) ∩ ∂M = ∅ and the jets of f are transverse to

n−1,0

, Σ

n,0

, Σ

n−1,1,0

, Σ

n−1,1,1,0

, S

and S

on the part corresponding

to f

−1

(U).

Proof of Theorem 3.3. Set U = IntF . By Proposition 3.5 and Lemma 3.6,

we see that there exists a smooth map g : M → N such that g ∈ V, g = f

on f

−1

(U) = g

−1

(U) and that g is excellent on N − F . Since f and g are

continuous and N is Hausdorﬀ, we see that g = f on the closure of f

−1

(U).

However, we do not know if g is excellent on a neighborhood of N − IntF.

This is be cause there is a possibility of a point in f

−1

(N − F ) being mapped

into ∂F by g. In order to exclude this possibility, we modify the argument

as follows.

Since ∂F is a closed submanifold of N , the set of maps of M into N

transverse to ∂F forms an open set of C

∞

(M, N) (see [7, Proposition 4.5 (p.

52)]). Thus we may assume that every map in the open set V is transverse

to ∂F from the beginning. Furthermore, since the set of the proper maps

of M into N forms an open set (see [10, Theorem 1.5 (p. 38)]), we may

further assume that every element of V is a proper map. We may further

assume that each element of V maps ∂M into F − V by a similar reason,

where V is a closed neighborhood of ∂F in N . Now suppose that g ∈ V.

Then, since g

−1

(∂F ) is a closed regular submanifold of IntM, we see that

the closure of g

−1

(U) is equal to g

−1

(F ). Since f = g on the closure of

−1

(U) and f

−1

(U) = g

−1

(U), we see that f = g on f

−1

(F ) = g

−1

(F ).

Combining the facts that g is excellent on N − F and that f is excellent

on a neighborhood of ∂F , we see that g is excellent on a neighborhood of

N − IntF . This completes the proof of Theorem 3.3. 

Remark 3.7. Results similar to Theorem 3.3 hold for some other dimension

pairs as well.

4. Graphic as the discriminant set.

Let Θ

, Θ

, H, G be as is Section 2, where H, G may not be

generic. In this section, we ﬁrst observe that we can obtain a smooth map

118 T. KOBAYASHI AND O. SAEKI

f : M − (Θ

∪ Θ

) → I × I from H and G, and we show, by

using Theorem 3.3, that f can be deformed to a map Φ which is excellent

in the exterior of a regular neighborhood of Θ

∪ Θ

by an

arbitrarily small deformation. Then we see that we can obtain sweep-outs

, G

associated to A ∪

B, X ∪

Y respectively from Φ, which have

the feature “generic” in Sect. 2. Finally we observe that the corresponding

graphic is actually the discriminant set (for the deﬁnition, see Sect. 3) of Φ.

Let M be a closed 3-manifold. In this section we consider a smo oth map

f from M to the Euclidean space R

. Recall that S(f) denotes the set of

singular points (or singular set) of f. That is,

S(f) = {q ∈ M| rank (df

: T

M → T

f(q)

) ≤ 1}.

Then as a special situation of Deﬁnition 3.2, we have:

Deﬁnition 4.1. Let f : M → R

be a smooth map. For an open set U of

, we say that f is excellent on U if f

−1

(U) ∩ ∂M = ∅, and the following

conditions are satisﬁed.

(1) For each point q ∈ S(f) there exist local coordinates (u, x, y) for q,

and (X, Y ) for f(q) such that:

(1-1) X ◦ f = u, Y ◦ f = x

+ y

, or

(1-2) X ◦ f = u, Y ◦ f = x

− y

, or

(1-3) X ◦ f = u, Y ◦ f = y

+ ux − x

(2) For each cusp q, f

−1

(f(q)) ∩ S(f) = {q} (that is, the ﬁber which

contains q does not contain another singular point).

(3) f|

S(f )−{cusps}

is an immersion (possibly) with normal crossing (that

is, an immersion (possibly) with transverse self intersections).

We call a singular point of type (1-1) ((1-2) resp.) a deﬁnite fold (indeﬁnite

fold resp.). Recall that a singular point of type (1-3) is c alled a cusp.

Figure 4.1.

RUBINSTEIN–SCHARLEMANN GRAPHIC 119

Figure 4.2.

Figure 4.3.

Figure 4.4.

Now we describe the relationship between graphic and excellent map. Let

M, A ∪

B, X ∪

Y , Θ

, Θ

, and Θ

be as in Section 2. (Here we

suppose that Θ

∪ Θ

and Θ

∪ Θ

are in general position.)

Let H, G be sweep-outs obtained from the Heegaard splittings A ∪

X ∪

Y respectively. We may suppose that H|

P ×(0,1)

: P × (0, 1) → M −

(Θ

∪ Θ

) and G|

Q×(0,1)

: Q × (0, 1) → M − (Θ

∪ Θ

) are smooth. Let

Φ : M − (Θ

∪ Θ

) → I × I be the map deﬁned by:

(∗) Φ(p) = (s, t) ⇐⇒ p ∈ P

∩ Q

Since H|

P ×(0,1)

, G|

Q×(0,1)

are smooth maps, we see that Φ is also smooth.

Since {P

} and Θ

∪ Θ

, {Q

} and Θ

∪ Θ

are generic, we see that:

120 T. KOBAYASHI AND O. SAEKI

(∗1) H (G resp.) is standard (see Sect. 2) in a small regular

neighborhood N(Θ

∪Θ

) of Θ

∪Θ

(N(Θ

∪Θ

) of Θ

∪Θ

resp.), and, hence, Φ is transverse to the frontier of a regular

neighborhood of ∂(I × I) in I × I.

By Theorem 3.3, we see that we can deform Φ in M − (N(Θ

∪ Θ

) ∪

N(Θ

∪ Θ

)) by an arbitrarily small deformation, to a map Φ

which is

excellent on the complement of the regular neighborhood of ∂(I × I) in

I × I, and this together with (∗1) implies that Φ

is excellent on Int(I × I).

Since Φ

is obtained from Φ by a small deformation, we may suppose:

(∗2) pr

◦ Φ

, pr

◦ Φ

have no critical points, where pr

, pr

I × I → I are the projections to the ﬁrst, and second factors

respectively.

By condition (∗2), we see that, there exist sweep-outs H

, G

such that

(P × {s}) = Φ

0−1

({s} × I), and G

(Q × {t}) = Φ

0−1

(I × {t}). Note that

, G

are small deformations of H, G. By the deﬁnition of H

, G

, we

immediately have:

(∗)

(p) = (s, t) ⇐⇒ p ∈ H

(P × {s}) ∩ G

(Q × {t}).

Then by Deﬁnition 4.1 (and the deﬁnition of the graphic in Sect. 2) we se e

that H

and G

are generic in the sense of Rubinstein and Scharlemann (see

Sect. 2) and, by comparing Deﬁnition 4.1 and the deﬁnition of the graphic

in Sect. 2, it is easy to see that the corresponding graphic is actually the

image of the singular set of Φ

on M − (Θ

∪ Θ

), where the

image of a deﬁnite fold corresponds to a center tangency, the image of an

indeﬁnite fold corresponds to a saddle tangency, and the image of a cusp

corresponds to a birth-death vertex.

Now we summarize the above results to give:

Theorem 4.2. Let H, G be as above. Then, by an arbitrarily small defor-

mation of H and G, we obtain sweep-outs H

and G

such that:

1. The above map Φ

(see (∗)

) is excellent on Int (I × I),

2. The maps H

and G

are generic. Hence we can obtain a graphic Γ

from H

and G

, and then Γ ∩ Int (I × I) is the discriminant set of the

excellent map Φ

M−(Θ

∪Θ

)

5. Isotopy, and stabilization of Heegaard surfaces.

Let f : M → I × I(⊂ R

) be an excellent map obtained in the previous

section (which is denoted by Φ

there). Let α : I → R

be an embedding of

the unit interval.

Deﬁnition 5.1. We say that α is transverse to f if α satisﬁes the following

two conditions:

RUBINSTEIN–SCHARLEMANN GRAPHIC 121

(1) α(∂I) ⊂ ∂(I × I), and α and ∂(I × I) are transverse (i.e., a smooth

slight extension of α is transverse to smooth extensions of I × {0}, I × {1},

{0} × I, and {0} × I),

(2) for each pair (t, q) ∈ (0, 1) × M with α(t) = f(q), we have:

M) + dα

I) = T

f(q)

Then, by Deﬁnition 4.1, it is easy to see:

Lemma 5.2. Suppose that α satisﬁes Deﬁnition 5.1 (1). Then α is trans-

verse to f if and only if:

For q ∈ M with f(q) ∈ α(I), w e have either one of:

1. q is not a singular point of f,

2. q is a fold point which is not a normal crossing, and α is transverse

to the discriminant set at f(q),

3. q is a fold point which is mapped to a normal crossing, and α is trans-

verse to the two arcs (which are local images of the singular set), or

4. q is a cusp. In this case, there are two arcs in a neighborhood of

f(q) (which are local images of the singular set). Then there is a slight

smooth extensions of the arcs, say `

, `

such that α is transverse to

, and `

Figure 5.1.

Lemma 5.3. Suppose that α is transverse to f. Then f

−1

(α(I)) is a 2-

dimensional submanifold of M.

Proof. By condition Deﬁnition 5.1 (2), we see that f

−1

(α(0, 1)) is a 2-

dimensional proper s ubmanifold in M − (Θ

∪ Θ

). Then by

condition Deﬁnition 5.1 (1), we see that f

−1

(α[0, ε]) (f

−1

(α[1 − ε, 1]) resp.)

is a disk, which cap oﬀ f

−1

(α(0, 1)) to make a closed surface f

−1

(α(I)). 

Proposition 5.4. Let α, β be arcs transverse to f. Suppose that α(I) is

deformed to β(I) through a sequence of moves of the following types.

(0) Ambient isotopy of I × I wh ich ﬁxes ∂(I × I) ∪ Γ setwise.

(1) Passing a crossing vertex as α

−

→ α

in Figure 5.2.

122 T. KOBAYASHI AND O. SAEKI

Figure 5.2.

(2) Passing a cusp as α

−

→ α

in Figure 5.3.

Figure 5.3.

(3) Passing a vertex in ∂(I × I) as in Figure 5.4.

Figure 5.4.

(4) Passing a corner of ∂(I × I) as in Figure 5.5.

Figure 5.5.

Then the surfaces f

−1

(α(I)), and f

−1

(β(I)) are isotopic in M.

Proof. First, we consider moves (0), (1), and (2). Suppose that α

(I) is

deformed to α

(I) through a sequence of moves (0), (1), and (2). By

Lemma 5.2, we see that there is a 1-parameter family of transverse arcs

RUBINSTEIN–SCHARLEMANN GRAPHIC 123

(0 ≤ s ≤ 1) from α

to α

. Then we obtain an isotopy of surfaces

−1

(α

) in M.

Now we consider move (3). Note that the deformation (3) gives the isotopy

as in Figure 5.6.

Figure 5.6.

This shows that the deformation (3) gives mutually isotopic surfaces.

Finally we consider about move (4). Note that in a neighborhood of a

corner of I × I, P

and Q

are disjoint. Hence the deformation (4) obviously

gives equal surfaces.

Combining the above observations, we have the conclusion of the propo-

sition. 

As a consequence of Propos ition 5.4. we have:

Corollary 5.5 (cf. Example 2.10). Suppose t hat the graphic obtained from

P and Q contains a region as in Figure 5.7. Then P and Q are isotopic in

Figure 5.7.

Proof. Let α(t) = (t, ε), and β(t) = (ε, t) for a small ε > 0. It is easy to

see that α(I) is deformed to β(I) within the above region by applying the

deformations of Proposition 5.4. 

For a stabilization of a Heegaard splitting, we have:

Proposition 5.6. Let α be an arc transverse to f such that f

−1

(α(I)) is a

Heegaard surface. Suppose that a transverse arc α

is obtained by changing

124 T. KOBAYASHI AND O. SAEKI

α locally as in Figure 5.8. Then f

−1

(α

(I)) is a Heegaard surface which is

a stabilization of Φ

−1

(α(I)).

Figure 5.8.

Proof. We use the following picture (Figure 5.9) for the proof. The picture

corresponds to the point of the intersection of α(I) and the image of an

indeﬁnite fold (here P

’s are represented by horizontal planes).

Figure 5.9.

Let P

be the subsurface of Φ

−1

(α

(I)) corresponding to Φ

−1

(α

), where

is as in Figure 5.8. It is directly observed from Figure 5.9 that each P

looks as in Figure 5.10.

Figure 5.10.

By summing up P

’s, we see that Φ

−1

(α

(I)) is a stabilization of

−1

(α(I)). 

Corollary 5.7. Let P , and Q be the Heegaard surfaces as in Example 2.11.

Then P and Q become isotopic by applying one stabilization.

RUBINSTEIN–SCHARLEMANN GRAPHIC 125

Proof. We ﬁrst take transverse arcs α, and β as in Figure 5.11. Then, by

Figure 5.11, we see that α(I) can be deformed to the arc in Figure 5.12,

by one application of the deformation of Proposition 5.6, and deformations

in Proposition 5.4. By reﬂecting the pictures in Figure 5.11 in the line

connecting right-bottom corner to left-top corner, we see that β(I) is also

deformed to the arc in Figure 5.12, and this gives the conclusion. 

Figure 5.11.

Figure 5.12.

6. Orbifold version of Rubinstein-Scharlemann graphic.

In this section, we formulate an orbifold version of the Rubinstein-Scharle-

mann setting, and show that the local lab e lling scheme described in [ 18]

holds in this setting.

Let M be a compact 3-manifold, γ a union of mutually disjoint arcs or

simple close d curves properly embedded in M, F a surface properly em-

bedded in M, which is in general position with respect to γ, and `(⊂ F ) a

simple closed curve with ` ∩ γ = ∅.

Deﬁnition 6.1. A surface D is a γ-disk, if D is a disk intersecting γ in at

most one transverse point.

Deﬁnition 6.2. We say that ` is γ-inessential if ` bounds a γ-disk in F .

We say that ` is γ-essential if it is not γ-inessential.

126 T. KOBAYASHI AND O. SAEKI

Let `

, `

(⊂ F) be simple closed curves with `

∩ γ = ∅ (i = 1, 2).

Deﬁnition 6.3. We say that `

and `

are γ-parallel if `

∪ `

bounds an

annulus A in F such that A ∩ γ = ∅.

Deﬁnition 6.4. We say that D is a γ-compressing disk for F if; D is a

γ-disk; and D ∩ F = ∂D, and ∂D is γ-essential in F . The surface F is

γ-compressible if it admits a γ-compressing disk, and it is γ-incompressible

if it is not γ-compressible.

Let a be an arc properly embedded in F with a ∩ γ = ∅.

Deﬁnition 6.5. We say that a is γ-inessential if there is a subarc b of ∂F

such that ∂b = ∂a, and a ∪ b bounds a disk D in F such that D ∩ γ = ∅.

We say that a is weakly γ-inessential if there is a subarc b of ∂F such that

∂b = ∂a, and a ∪ b bounds a γ-disk D in F .

Deﬁnition 6.6. Let F

, F

be surfaces embedded in M such that ∂F

∂F

. We say that F

and F

are γ-parallel, if there is a submanifold N in M

such that (N, F

∩F

, N ∩γ) is homeomorphic to (F

×I, ∂F

×{1/2}, P ×I)

as a triple, where P is a union of points in Int(F

), and F

resp.)

corresponds to the closure of the component of ∂(F

× I) − ∂F

× {1/2}

containing F

× {0} (F

× {1} resp.).

The submanifold N is called a γ-parallelism between F

and F

We say that F is γ-boundary parallel if there is a subsurface F

in ∂M

such that F and F

are γ-parallel.

Deﬁnition 6.7. Let F

, F

be mutually disjoint surfaces in M which are

in general position with respect to γ. We say that F

and F

are γ-isotopic

if there is an ambient isotopy φ

(0 ≤ t ≤ 1) of M such that; φ

= id

;

) = F

, and; φ

(γ) = γ for each t.

Genus g n-bridge position.

Let Γ = {γ

, . . . , γ

} be a system of mutually disjoint arcs properly em-

bedded in M.

Deﬁnition 6.8. We say that Γ is trivial if there exists a system of mutually

disjoint disks {D

, . . . , D

} in M such that (1) D

∩ Γ = ∂D

∩ γ

= γ

, and

(2) D

∩ ∂M is an arc, say α

, such that α

= c`(∂D

− γ

Example 6.9. Let β be a system of trivial two arcs in a 3-ball B. The pair

(B, β) is often refered as 2-string trivial tangle, or a rational tangle.

Let K be a link in a closed 3-manifold M. Let M = A ∪

B be a genus

g Heegaard splitting. Then the next deﬁnition is borrowed from [6].

Deﬁnition 6.10. We say that K is in a (genus g) n-bridge position (with

respect to the Heegaard splitting A ∪

B) if K ∩A (K ∩ B resp.) is a system

of trivial n arcs in A (B resp.).

RUBINSTEIN–SCHARLEMANN GRAPHIC 127

In this paper, we abbreviate genus 0 n-bridge position to n-bridge posi-

tion.

Unknotting tunnel.

Let K be a knot in a closed 3-manifold M . A tunnel for K is an embedded

arc σ in S

such that σ∩K = ∂σ. We say that a tunnel σ for K is unknotting

if S

−Int N (K ∪ σ, S

) is a genus two handlebody.

Orbifold setting.

Let K be a link in a closed 3-manifold M. We regard K as γ above. Let

, L

be a pair of mutually disjoint 1-complexes in M such that:

1. Each vertex of L

has valency zero, one or three,

2. (L

∪ L

) ∩ K consists of the union of a (possibly, empty) sublink of

K, and a subset of the vertices of L

∪ L

with valency one,

3. Let N be a regular neighborhood of L

∪ L

, and E = c`(M − N).

Then (E, E ∩ K) is homeomorphic to (P × (0, 1), P × (0, 1)), where P

is a closed surface, and P is a ﬁnite set of (possibly empty) points in

P .

Let A, B be the closures of the components of M − (P × {1/2}), where

⊂ A, L

⊂ B. We say that A ∪ B is an orbifold Heegaard splitting

of (M, K). Then as in Sect. 2, we obtain a sweep-out H : P × I → M.

Let R

, R

be another pair of 1-complexes satisfying the above conditions

(1), (2), and (3), and G : Q × I → M the corresponding sweep-out. Then,

as in Theorem 4.2, we may suppose that we can obtain an excellent map

f : M − (L

∪ L

∪ R

) → R

from H and G such that the graphic

obtained from H and G is the discriminant set f(S(f)) ⊂ I × I. Here we

note that we have to slightly generalize the deﬁnition of standard position

for a neighborhood of a valency one vertex (e.g. Figure 6.1), and it is easy

to see the procedures in Sect. 4 work under this situation.

Figure 6.1.

Example 6.11. Suppose that K is in a genus g n-bridge position with

respect to a Heegaard splitting A ∪

B. Then, by adding n edges to each

of the appropriate spines of A and B, we can obtain 1-complexes satisfying

the above conditions.

128 T. KOBAYASHI AND O. SAEKI

Figure 6.2.

Example 6.12. Let K be a tunnel number one knot with unknotting tunnel

τ. Let L

= K ∪ τ. Let L

be a spine of the genus two handlebody c`(M −

N(K ∪ τ)) with each vertex having valency three. Then L

, L

satisﬁes the

above conditions.

Deﬁnition 6.13. Let H, G be sweep-outs as above. We say that H and G

are K-comparable if f|

K−(L

∪L

∪R

)

: K−(L

∪L

∪R

) → Int(I×I)

is an immersion (possibly) with normal crossing, and f(K − (L

∪ L

∪ R

∪

)) and f(S(f)) are in general position in Int(I × I).

Proposition 6.14. By an arbitrarily small deformation on K rel (L

∪L

∪

∪ R

) with respect to Whitney topology, we can arrange H and G to be

K-comparable.

Proof. Note that f

−1

(f(S(f ))) has the structure of a simplicial complex with

dimension at most 2 (see Figure 6.3), and, hence, by an arbitrarily small

deformation of K with respect to Whitney topology we can arrange so that

K and f

−1

(f(S(f ))) are in general position, that is, K and the 1-skeleton are

disjoint, and K and f

−1

(f(S(f ))) intersect transversely in a ﬁnite number

of points. This shows that f (K) and f(S(f)) intersect transversely in a

ﬁnite number of points. Possibly f(K) may contain a crossing vertex of the

graphic f(S(f)). Then we further apply a small deformation to make f(K)

avoid crossing vertices and to make f|

K−(L

∪L

∪R

)

an immersion with

normal crossing, and this gives the conclusion. 

Figure 6.3.

RUBINSTEIN–SCHARLEMANN GRAPHIC 129

For a K-comparable pair H and G we can obtain a graphic Γ as in the

following manner.

Regions: A region is a component of the subset of Int(I × I) consisting

of values (s, t) such that P

and Q

intersect transversely, and K ∩

∩ Q

) = ∅.

Edges: An edge is a component of the subset consisting of values (s, t)

such that either:

(1) P

and Q

intersect transversely except for one non-degenerate

tangent point and K ∩ (P

∩ Q

) = ∅, or

(2) P

and Q

intersect transversely and K ∩ (P

∩ Q

) consists of

one point.

By Deﬁnition 6.13, we see that edge is a 1-dimensional subset of

Int(I × I) which is monotonously increasing or decreasing.

Crossing vertices: A crossing vertex is a com ponent of the subset con-

sisting of points (s, t) such that either:

(1) P

and Q

intersect transversely except for two non-degenerate

points of tangency and K ∩ (P

∩ Q

) = ∅, or

(2) P

and Q

intersect transversely except for one non-degenerate

tangent point and K ∩ (P

∩ Q

) consists of one point, or

(3) P

and Q

intersect transversely and K ∩ (P

∩ Q

) consists of

two points.

Note that in this setting we may (as in Sect. 2) also regard a cross-

ing vertex to be a crossing point of two edges. This follows from

the same reason as in Section 2 (Case (1)), or from the condition

“K and f

−1

(f(S(f ))) are generic” (Case (2)), or from the condition

“f|

K−(L

∪L

∪R

)

is an immersion with normal crossings” (Case (3)).

Birth-death vertices: A birth-death vertex is a c omponent of the sub-

set consisting of points (s, t) such that P

and Q

intersect transversely

except for a single degenerate tangent point and K ∩ (P

∩ Q

) = ∅.

Labelling regions of the graphic.

Consider a region of the graphic I × I − Γ. Then the K-isotopy class of

∩ Q

in P

or Q

is independent of the choice of (s, t) in each region, and,

hence, we often abbreviate P

by P , and Q

by Q.

The purp ose of the rest of this section is to claim that the nature of the

(local) labelling schemes discussed in [18] holds also in our setting. We

assume that the reader is familiar with Sect. 4, 5 of [18].

Deﬁnition 6.15. We say that an orbifold Heegaard splitting A ∪

B is

weakly K-reducible if there exist K-compressing disks D

, D

for P in A,

B respectively such that ∂D

∩ ∂D

= ∅. The orbifold Heegaard splitting

A ∪

B is strongly K-irreducible if it is not weakly K-reducible.

130 T. KOBAYASHI AND O. SAEKI

Deﬁnition 6.16. Let (s, t) be a point in a region of I ×I −Γ. (Hence, P ∩Q

consists of a system of s imple closed curves in M disjoint from K.) Let C

resp.) be the subset of the simple closed curves which are K-essential

in P (Q res p.). Then the subset C

of C

is deﬁned by:

= {c|c bounds a K-disk D in Q − C

such that N(∂D, D) ⊂ A},

where N(∂D, D) is a regular neighborhood of ∂D in D.

Analogously C

(⊂ C

), and C

, C

(⊂ C

) are deﬁned.

Lemma 6.17 (Lemma 4.3 of [18]). If c ∈ C

, then c bounds a K-disk in

Proof. The proof is basically the same as Rubinstein-Scharlemann’s except

for the consideration on K. That is:

Let D be the K-disk which c bounds in Q, such that N (∂D, D) ⊂ A. If

IntD ∩ P = ∅, then D gives a desired K-disk. Suppose that IntD ∩ P 6= ∅.

Let ∆(⊂ D) be an innermost disk. Since IntD ∩ C

= ∅, we see that ∂∆

bounds a K-disk ∆

in P . For a proof of the next claim, see Appendix A-3.

Claim. ∆

∩ K = ∅ if and only if ∆ ∩ K = ∅. Furthermore, if ∆

∩ K 6= ∅,

then ∆ and ∆

are K-parallel, i.e., ∆ ∪ ∆

bounds a 3-ball D

such that

∩ K is an unknotted arc.

By the claim, we see that we can apply cut and paste on D using ∆ and

∆

to get a new disk D

with fewer intersections. By applying the argument

ﬁnitely many times, we obtain the desired disk. 

As an immediate consequence of Lemma 6.17, we have:

Corollary 6.18 (Corollary 4.4 of [18]). If there exists a region such that

both C

and C

are non-empty, then A ∪

B is weakly K-reducible.

In the rest of this section, we suppose:

M admits a 2-fold branched covering space p :

M → M along

Lemma 6.19 (Lemma 4.5 of [18]). Suppose that C

= ∅, C

= ∅, and

there exists a ∂-reducing K-disk in A which intersects Q only in K-inessen-

tial simple closed curves. Suppose, moreover, that A contains a K-essential

curve of Q. Then either A ∪

B is weakly K-reducible, or M is the 3-sphere

and K is a trivial knot.

Proof. By Appendix A-3, we may suppose, by K-isotopy, that P and Q are

disjoint, and that the ∂-reducing K-disk D and Q are disjoint. Without loss

of generality, we may suppose that Y is contained in A. Now consider the

2-fold branched covers (along K)

P ,

Y ,

D of A, B, P , X, Y , D

respectively. Note that, by the deﬁnition of an orbifold Heegaard splitting,

B are handlebodies.

RUBINSTEIN–SCHARLEMANN GRAPHIC 131

Take a maximal compression body

C of

A − Int

Y for ∂

A. By the unique-

ness of maximal compres sion body, we may suppose, by applying Z

-equi-

variant cut and paste arguments as in the Proof of 10.3 of [9] or [12], that

C is invariant under the cove ring translation τ . Let P

be a component of

the inner boundary of

If P

is a sphere, then M is S

(see Proof of [18, Lemma 4.5]), and, by

-Smith Conjecture ([21], or [16]), K is a trivial knot in S

Suppose that P

is not a sphere. Note that P

is compressible in

B ∪

since P

is contained in a handlebody

X (see the proof of Lemma 4.5 of

[18]). Then note that

B ∪

C is a Heegaard splitting in the sense of Casson-

Gordon [4]. Hence, by [4], there exists a compressing disk D

(⊂

B ∪

C) for

such that D

∩

P consis ts of a circle (hence, D

∩

C is an annulus).

Now we show that we can have such D

which moreover is equivariant

with respect to τ. We may suppose, by general position argument, that D

and τ(D

) intersect transversely. Then, by isotopy, we may suppose that

each component of

C ∩ (D

∩τ(D

)) is an essential arc in the annulus D

∩

(and τ(D

) ∩

C). Then, by applying equivariant cut and paste arguments as

in the proof of 10.3 of [9] or [12] on D

, and τ(D

), we obtain an equivariant

compressing disk(s) D

. Since each component of

C ∩ (D

∩ τ (D

)) is an

essential arc of D

∩

C, we see that each component of D

intersects

C in

an annulus, (hence, intersects

P in a circle).

Then apply cut and paste arguments on D

∩

C and

D to obtain a com-

pressing disk

(⊂

C) for

P such that

∩ (D

∩

C) = ∅. Then, by

applying equivariant cut and paste arguments on

, and τ(

), we obtain

equivariant disk(s) D

(⊂

C) for

P such that D

∩ D

= ∅. Then p(D

and p(D

∩

B) give weak K-reducibility of K. 

Labelling scheme.

Now we mimic the procedures in [18, Section 5]. If C

, C

resp.) is non-empty, then we label the region A (B, X, Y resp.). If C

and

are both empty and A (B resp.) contains an K-essential curve of Q,

then we lab el the region b (a resp.), and if X (Y resp.) contains an essential

curve of P , then we label the region y (x resp.). By Corollary 6.18 we have:

Rule 1. If there exists a region with both labels A and B assigned, then

A ∪

B is weakly K-reducible.

We obviously have:

Rule 2. No region can have both an upper case label and lower case label.

Next, we consider how labels change as one cross an edge of Γ.

Note that we have the following three pos sibilities.

1) The edge comes from center tangency.

132 T. KOBAYASHI AND O. SAEKI

In this case, the regions have exactly the same label.

2) The edge comes from saddle tangency.

In this case, the eﬀect is banding two components of P ∩ Q, say c

and

, to make a simple closed curve, say c, or vice versa.

3) The edge comes from P ∩ Q ∩ K.

In this case the eﬀect is that a component of P ∩ Q passes a puncture by

K on P, (and Q).

Note that situation 3) did not appear in Rubinstein-Scharlemann setting.

With this fact in mind, it is easy, by tracing the proof of [18, Corollary 5.1],

to see:

Rule 3 ([18, Corollary 5.1]). If both labels A and B appear in two adjacent

regions, then A ∪

B is weakly K-reducible.

Then we have:

Rule 4 ([18, Corollary 5.2]). In adjacent regions of I × I − Γ, labels a and

b (x and y resp.) cannot app ear.

Proof. Suppose that a and b occur opposite sides of an edge. Then argu-

ments in the proof of [18, Corollary 5.2] show that edge does not come from

saddle tangency. Then it is easy to see that this phenomena can occur only

in case when Q is a 2-sphere and Q∩K consists of three points a

, a

and

a component of the intersection P ∩Q is changed from a circle separating a

and a

∪ a

to a circle separating a

∪ a

and a

. However this is impossible,

since Q ∩ K must consists of even number of points. 

With tracing the proof of [18, Lemma 5.3] with consideration on K we

easily have:

Lemma 6.20 ([18, Le mma 5.3]). Suppose, in I × I − Γ, a region labelled

A is adjacent to a region labelled with a lower case letter. Then the edge

represents either (1) a saddle tangency in which a band which is K-essential

in P and weakly K-inessential in Q is attached to an intersection curve

which is K-inessential in both P and Q, or (2) a passing of K which changes

an element of C

bounding a disk (in P) with two punctures by K into a

disk with one puncture by K.

Then we have:

Rule 5 ([18, Corollary 5.4]). Suppose, in I × I − Γ, a region labelled A is

adjacent to a region labelled b. Then either A ∪

B is weakly K-reducible,

or M

∼

and K is a trivial knot.

Proof. We see, by Lemma 6.20, that A ∪

B satisﬁes the assumption of

Lemma 6.19, and this gives the conclusion. 

RUBINSTEIN–SCHARLEMANN GRAPHIC 133

In the following, the notation a stands for, as in [18], either a or A, and

similar for b, x, and y. With the above rules, we see that the arguments in

the proof of [18, Lemma 5.7] (it is easy to check that [18, Lemma 5.6] holds

in our setting since the new phenomenon is the situation 3) in the preceding

Rule 3) completely works in our setting to give:

Rule 6 ([18, Lemm a 5.7]). If all letters a, b, x, and y appear in quadrants

of a crossing vertex of Γ, then either two opposite quadrants are unlabelled,

or one of A ∪

B, X ∪

Y is weakly K-reducible, or M

∼

and K is a

trivial knot.

By using these rules, the arguments in the proof of [18, Proposition 5.9]

show (the diﬀerence here is the consideration on K, a possibility that three

edges may be joined to a vertex in ∂(I × I) (see Figure 6.1)):

Proposition 6.21. Let A ∪

B, X ∪

Y be orbifold Heegaard splittings for

(M, K) obtained from two bridge positions as in Example 6.11. Suppose that

A ∪

B, X ∪

Y are strongly K-irreducible, and K is not a trivial knot in

. Then there is an unlabelled region in I × I − Γ.

And, this together with Appendix A-3, and the arguments in the proof

of [18, Corollary 6.2] shows:

Corollary 6.22. Let A ∪

B, X ∪

Y be as in Proposition 6.21. Then,

by applying K-isotopy, we may suppose that P and Q intersect non-empty

collection of simple closed curves which are K-essential in both P and Q.

7. 2-bridge position of a 2-bridge knot.

Let K be a non-trivial 2-bridge knot (that is, K is a non-trivial knot which

admits a genus 0 2-bridge position). In this section, we show that the

2-bridge positions of K are unique up to K-isotopy, which was originally

proved by Schubert [20].

Theorem 7.1. Let K be a non-trivial 2-bridge knot, and P , Q are 2-spheres

in S

which give 2-bridge positions of K. Then P is K-isotopic to Q, i.e.,

there is an ambient isotopy ϕ

(0 ≤ t ≤ 1) of S

such that (1) ϕ

(K) = K

(0 ≤ t ≤ 1), (2) ϕ

= id

, and (3) ϕ

(P ) = Q.

For the proof of Theorem 7.1, we prepare some lemmas, proofs of which

are given in Appendix B. (For the deﬁition of β-essential surface, see Deﬁ-

nition 6.2.)

Lemma 7.2 (Appendix B-1). Let (B, β) be a 2-string trivial tangle. Let F

be a surface properly embedded in B. Suppose that F is β-essential. Then

F is a disk which is disjoint from β, and F separates the components of β.

134 T. KOBAYASHI AND O. SAEKI

Figure 7.1.

Recall that it is often said that (C, γ) is a rational tangle if (C, γ) is

homeomorphic to the 2-string trivial tangle (B, β) as a pair.

Lemma 7.3 (Appendix B-2). Let (B, β) be a 2-string trivial tangle, and F

a β-incompressible surface in B.

Then either (0) F is β-essential (see Lemma 7.2), (1) F is a β-boundary

parallel disk intersecting β in at most one point, (2) F is a β-boundary

parallel disk intersecting β in two points (and, hence, F separates (B, β)

into the parallelism and a rational tangle), or (3) F is β boundary parallel

annulus such that F ∩ β = ∅.

Figure 7.2.

Lemma 7.4 (Appendix B-3). Let D be a β-compressible disk in B such

that ∂D is β-essential in ∂B, and D ∩ β consists of two points. Then D

separates (B, β) into two tangles (B

, β

), and (B

, β

), where (B

, β

) is

a rational tangle such that there is a β-essential disk D

in (B

, β

) with

D ∩ D

= ∅. Moreover if (B

, β

) happens to be a rational tangle, then

, β

) is a β-boundary parallelism for D.

Figure 7.3.

RUBINSTEIN–SCHARLEMANN GRAPHIC 135

Let A, B (X, Y resp.) be the closures of the components of S

−P (S

−Q

resp.).

Proposition 7.5. Every genus 0 Heegaard splitting of S

which gives a 2-

bridge position of K is strongly K-irreducible.

Proof. We give the proof for A ∪

B. Assume that A ∪

B is weakly K-

reducible, and let D

, D

be a pair of K-esssential disks in A, B respectively

such that ∂D

∩ ∂D

= ∅. Since P ∩ K consists of four points, we see that

∂D

and ∂D

are parallel in P − K, and this together with Lemma 7.2

implies that K is a 2-component trivial link, a contradiction. 

Proof of Theorem 7.1. Note that, by Proposition 7.5, A ∪

B, X ∪

Y are

strongly K-irreducible. Hence , by Corollary 6.22, we may suppose that

P and Q intersect non-empty collection of simple closed curve s which are

K-essential in both P and Q.

The proof is carried out by the induction on the number of the components

of P ∩ Q. The following Claims 1 and 2 give the ﬁrst step of the induction.

Claim 1. If P ∩ Q consists of one component, then P and Q are K-isotopic.

Proof. Let D

= Q∩A, D

= Q∩B, D

= P ∩X, and D

= P ∩Y . Then

, D

are disks each of which intersects K in two points. Then

we have the following case s.

Case 1. The disks D

, D

are K-incompressible in A, B, X, Y

respectively.

In this case, by Lemma 7.3 (2), we have:

(1) “D

and D

are K-parallel in A” or “D

and D

are K-parallel in A”,

(2) “D

and D

are K-parallel in B” or “D

and D

are K-parallel in B”,

(3) “D

and D

are K-parallel in X” or “D

and D

are K-parallel in

X”, and

(4) “D

and D

are K-parallel in Y ” or “D

and D

are K-parallel in Y ”.

It is easy to see that the above 4 conditions imply either one of:

(1) “D

and D

are K-parallel in A (and, X), and D

and D

are K-

parallel in B (and, Y )”, or

(2) “D

and D

are K-parallel in A (and, Y ), and D

and D

are K-

parallel in B (and, X)”.

Since the argument is symmetric, we may suppose that (1) holds. Then,

by using the parallelisms, we can move D

to D

, and D

to D

to give a

desired K-isotopy.

Case 2. One of the disks D

, D

, or D

is K-compressible in A, B,

X, or Y .

136 T. KOBAYASHI AND O. SAEKI

Without loss of generality, we may suppose that D

is K-compressible in

Y , and the K-compressing disk is contained in B. Note that this implies that

is K-compressible in B. Note moreover that D

is K-incompressible

in X, since D

separates the boundary components of each component of

K∩X in X, and similarly D

is K-incompressible in A. Then, by Lemma 7.3

(2) and the last half of Lemma 7.4, we see that D

and D

are K-parallel

in B (and, X). Similarly we can show that D

and D

are K-parallel in A

(and, Y ). Hence we can obtain a desired K-isotopy as in Case 1.

This completes the proof of Claim 1. 

Claim 2. If P ∩Q consists of two components, then P and Q are K-isotopic.

Proof. Let D

, A

, D

be the closures of the components of P −(P ∩Q) such

that D

and D

are disks, and A

is an annulus. Without loss of generality,

we may suppose that D

∪ D

is contained in X, and A

is contained in Y .

Subclaim. Either D

or D

is K-boundary parallel in X.

Proof. If D

or D

is K-incompressible, then this immediately follows from

Lemma 7.3 (2). Suppose that D

and D

are K-compressible. Let B

be the

closure of the component of X − D

which corresponds to B

in Lemma 7.4.

By exchanging suﬃx, if necessary, we may suppose that Int(B

) ∩ P =

∅. Without loss of generality, we may suppose that B

is contained in A.

Let D

= B

∩ Q. Since D

separates the boundary components of each

component of K ∩ A in A, we see that D

is K-essential in A. Hence,

by Lemma 7.3 (2) and the last half of Lemma 7.4, we see that B

is a

K-parallelism.

Let B

be the parallelism between D

and ∂X obtained in Subclaim.

Then we can push D

out of X along the parallelism, and we have the

conclusion by Claim 1.

This completes the proof of Claim 2. 

Completion of Proof. Suppose that ](P ∩Q) ≥ 3. Note that the components

of P ∩ Q are mutually K-parallel in P . Let D

, A

, . . . , A

, D

be the

closures of the components of P − (P ∩ Q) such that D

, D

are disks and

, . . . , A

are annuli that are located on P in this order. Suppose that

or D

is K-boundary parallel in X or Y . Then, by using the arguments

in the proof of Claim 2, we can reduce ](P ∩ Q), to give the conclusion.

Suppose that D

and D

are not K-boundary parallel in X and Y . By

Lemma 7.3 (2), this implies that D

and D

are K-compressible.

Claim 3. Both D

and D

are contained in the closure of a component of

−Q, say X. And each component of P ∩Y is a K-incompressible annulus.

RUBINSTEIN–SCHARLEMANN GRAPHIC 137

Proof. Ass ume that D

is contained in X, and D

is contained in Y . By

applying K-compression on D

resp.) we obtain a K-essential disk E

resp.) in X (Y resp.) such that ∂E

= ∂D

. Note that E

resp.)

separates the components of K ∩ X (K ∩ Y resp.). This implies that K

is a 2-component trivial link, a contradiction. Hence we may suppose that

and D

are contained in X. Let A

be a component of P ∩ Y . Assume

that A

is K-compressible in Y . Then, by applying K-compression on A

we obtain two K-essential disks in Y . Then, by the above argument, we see

that K is a 2-c omponent trivial link, a contradiction.

Claim 3 together w ith Lemma 7.3 (3) implies that each component of

P ∩ Y is a K-boundary parallel annulus in Y . Take an outermost one of

P ∩ Y , say A

, and push A

out of Y along the parallelism. This reduces

](P ∩ Q) by two, and we have the conclusion by the assumption of the

induction.

This completes the proof of Theorem 7.1. 

8. Genus one 1-bridge position of a 2-bridge knot.

For a 2-bridge knot K we can obtain four genus one 1-bridge positions of K

as follows.

Let A ∪

B be the Heegaard splitting which gives the 2-bridge position.

Then let a

, a

, b

be the closures of the components of K − P , where

∪ a

∪ b

resp.) is contained in A (B resp.). Let T

= A ∪ N (b

, B),

= a

∪ b

∪ a

, T

= c`(B − N(b

, B)), and α

= b

. Then each T

is a

solid torus and α

is a trivial arc in T

, and, hence, T

∪ T

gives genus one

1-bridge position of K. Moreover, by using a

, a

, b

for b

, we can obtain

other three genus one 1-bridge positions of K.

Figure 8.1.

Remark 8.1. In [17], Morimoto-Sakuma study the isotopy classes of the

1-bridge positions ab ove.

138 T. KOBAYASHI AND O. SAEKI

We say that these genus one 1-bridge positions are standard.

The purpose of this section is to prove:

Theorem 8.2. Every genus one 1-bridge positions of a non-trivial 2-bridge

knot is standard.

First, we prepare some lemmas for the proof of Theorem 8.2, proofs of

which are given in Appendix C. Let α be a trivial arc in a solid torus T .

For (T, α), we have:

Lemma 8.3 (Appendix C-1). Let D be an α-compressing disk for ∂T .

Then D is either:

(1) a meridian disk of T with D∩α = ∅. In this case, we obtain, by cutting

(T, α) along D, a 1-string trivial tangle,

(2) a meridian disk of T with D ∩ α consists of one point, and we obtain,

by cutting (T, α) along D, a 2-string trivial tangle, o r

(3) ∂-parallel disk in T with D∩α = ∅. In this case, D cobounds a 1-string

trivial tangle in (T, α).

Figure 8.2.

Lemma 8.4 (Appendix C-2). Let D be an α-essential disk in T such that

D ∩ α consists of two points. Then there exists an α-compressing disk D

disjoint from D such that D

∩ α consists of one point. Moreover, by cutting

(T, α) along D

, we obtain 2-string trivial tangle (B, β) such that D is a

β-incompressible disk in (B, β) (hence D is β-boundary parallel).

We note that the disk D in Lemma 8.4 is either separating or non-

separating in T .

Figure 8.3.

Lemma 8.5 (Appendix C-3). Let D

, D

be mutually disjoint non α-paral-

lel, α-essential disks such that D

∩ α (i = 1, 2) consists of two points. Then

there exists an α-compressing disk D

for ∂T disjoint from D

∪ D

such

RUBINSTEIN–SCHARLEMANN GRAPHIC 139

that D

∩ α consists of one point. Moreover each D

is non-separating in T ,

and, by cutting (T, α) along D

, we obtain 2-string trivial tangle (B, β), and

, D

are mutually non β-parallel, β-boundary parallel, β-incompressible

disks in (B, β).

Figure 8.4.

Lemma 8.6 (Appendix C-4). Let D be an α-compressible disk such that

∂D is α-essential in ∂T , and D ∩ α consists of two points. Then there is a

disk ∆ in T such that ∆ ∩ D = ∂∆ ∩D = γ an arc, and ∆ ∩ α = c`(∂∆ − γ).

Particularly, if D is separating in T , then D separates (T, α) into (T

, α

and (B

, α

) such that α

is a trivial arc in a solid torus T

. In this case,

if (B

, α

) happens to be a rational tangle, then (B

, α

) is an α-boundary

parallelism.

Figure 8.5.

In the rest of this section, we give a proof of Theorem 8.2. Let K be a

non-trivial 2-bridge knot. Let A ∪

B be a genus 0 Heegaard splitting of

which gives a 2-bridge position of K, and X ∪

Y a genus one He egaard

splitting which gives a genus one 1-bridge position of K. Note that A ∪ B,

X ∪ Y give orbifold Heegaard splittings of (S

, K) (see Example 6.11).

Proposition 8.7. Exactly one of the following (1) or (2) holds.

1. X ∪

Y gives a standard genus one 1-bridge position.

2. X ∪

Y is strongly K-irreducible.

Proof. Suppose that X ∪

Y is weakly K-reducible. Let D

, D

be a pair

of K-esssential disks in X, Y respectively such that ∂D

∩ ∂D

= ∅. Since

) = {0}, we see that either D

is separating in X or D

is separating

in Y . Without loss of generality, we may suppose that D

is separating in

140 T. KOBAYASHI AND O. SAEKI

Claim 1. The disk D

is non-separating in Y .

Proof. Ass ume that both D

and D

are separating in X and Y respec-

tively. By Lemma 8.3 (3), we see that the closure of a component of X −D

(Y − D

resp.), say B

resp.), is a 3-ball such that K ∩ B

(K ∩ B

resp.) is a trivial arc . Since ∂D

∩ ∂D

= ∅, we see that ∂D

and ∂D

are K-parallel in Q. This implies that B

∪ B

is a 3-ball, and B

∩ B

a disk intersecting K in two p oints. This shows that K is a connected sum

of trivial knots, and, hence, K is a trivial knot, a contradiction. 

Claim 2. The disk D

intersects K in one point.

Proof. Ass ume this does not hold, i.e., D

∩K = ∅. Let N(D

) be a regular

neighborhood of D

in Y . Let X

= X ∪ N(D

), and Y

= c`(Y − N (D

)).

Then, by Lemma 8.3 (3), we see that X

is a 3-ball such that K ∩ X

a trivial arc. Moreover, by Lemma 8.3 (1), we see that Y

is a 3-ball such

that K ∩ Y

is a trivial arc. Hence we see that K is a trivial knot, a

contradiction. 

Let N

) be a regular neighborhood of D

in Y . Let X

= X∪N

and Y

= c`(Y − N

)). Then, by Lemma 8.3 (3), we see that X

is a

3-ball such that K ∩ X

is a system of 2-string trivial arcs. Moreover, by

Lemma 8.3 (2), we see that Y

is a 3-ball such that K ∩ Y

is a system of

2-string trivial arcs. Hence X

∪ Y

gives a 2-bridge position of K, and the

genus one 1-bridge position X ∪ Y is obtained from X

∪ Y

in a standard

manner.

Conversely, suppose that X ∪

Y gives a standard genus one 1- bridge

position. Then it is clear there exist disks corresponding to D

, D

above

in X, Y , and these disks give a weak K-irreducibility of X ∪

Y . This

together with the above shows that X ∪

Y gives a standard genus one

1-bridge position if and only if X ∪

Y is weakly K-irreducible.

This completes the proof of Proposition 8.7. 

Then we prove:

Proposition 8.8. Suppose tha t P ∩ Q consists of non-empty collection of

simple closed curves which are K-essential in both P and Q. Then the genus

one 1-bridge position X ∪

Y of K is obtained from A ∪

B in a standard

manner.

For the proof of Prop os ition 8.8, we prepare the following lemma, the

proof of which is left to the reader.

Lemma 8.9. Let T be a solid torus, and A an annulus properly embedded

in T such that each component of ∂A is a longitude of T . Then there is a

homeomorphism h : (annulus) × I → T such that h((annulus) × {1/2}) = A.

RUBINSTEIN–SCHARLEMANN GRAPHIC 141

Proof of Proposition 8.8. The proof is carried out by the induction of the

number of the components P ∩ Q. The following Claims 1 and 2 give the

ﬁrst step of the induction.

Claim 1. If P ∩ Q consists of one component, then the genus one 1-bridge

position X ∪

Y of K is obtained from A ∪

B in a standard manner.

Proof. Let D

= P ∩X, and D

= P ∩Y . Then D

, D

are disks properly

embedded in X, Y respectively, each of which intersect K in two points such

that ∂D

= ∂D

. Since P is separating in S

, D

is separating in X, and

this shows that ∂D

is contractible in Q. Let E be the disk in Q bounded

by ∂D

(= ∂D

). Then E intersects K in two points. Without loss of

generality, we may suppose that E is contained in A. Since E separates

the boundary components of each component of K ∩ A, we see that E is

K-incompressible in A. Then, by Lemma 7.3(2) we have either one of the

following two cases.

Case 1. E is not K-parallel to one of D

or D

in A.

In this case, we may suppose without loss of generality that E is not K-

parallel to D

. Then, by Lemma 7.3 (2), we see that E is K-parallel to D

in Y , and, by Lemma 8.4, we see that there is a K-compressing disk D for

∂X in X such that D intersects K in one point and D ∩ D

= ∅, and these

imply that the genus one 1-bridge position X ∪

Y of K is K-isotopic to a

genus one 1-bridge position which is obtained from A ∪

B in a standard

manner by using the arc K ∩ (B ∩ X).

Figure 8.6.

Case 2. E is K-parallel to D

and D

in A.

In this case, we consider the genus one surface F = Q ∩ B. Note that

∂F is a K-essential simple closed curve in P . Then, by Lemma 7.3, we see

that there is a K-compressing disk D for F in B. Without loss of generality,

we may suppose that D is contained in X. Then we have the following two

cases.

Case 2.1. D ∩ K = ∅.

In this case, we obtain a K-compressing disk D

for ∂B in B by com-

pressing F along D. By applying a slight isotopy, we may regard D

as a

142 T. KOBAYASHI AND O. SAEKI

K-compressing disk for ∂X in X such that ∂D

= ∂E. Then, by Lemma 8.6,

we s ee that the arc K ∩ X is pushed into by an isotopy rel ∂ in X to an arc

contained in E. Then we further push the arc along the parallelism between

E and D

to an arc contained in the disk D

. We denote by K

the image

of K under this isotopy. Then K

is contained in B and K

∩ ∂B is an arc.

Since this isotopy does not move the component of K ∩ B contained in Y

(= c`(K

∩ IntB)), c`(K

∩ IntB) is a trivial arc in B. This implies that K

is a trivial knot, a contradiction.

Figure 8.7.

Case 2.2. D ∩ K consists of one point.

In this case, we obtain, by applying K-compression on F along D and

slight isotopy, a disk D

in B with ∂D

= ∂E, and D

intersects K in two

points.

Subclaim. D

∪ E bounds a K-parallelism betwee n D

and E in X.

Proof. Ass ume not. Suppose that D

is K-compressible in X. Then, by

applying K-compression and slight isotopy, we obtain a K-compressing disk

in B such that ∂E

= ∂E. Then, by the argument of Case 2.1, we see that

K is a trivial knot, a contradiction. Suppose that D

is K-incompressible

in X. By the assumption, we see that E

∪ D

bounds a rational tangle in

X which is not a K-parallelism between E

and D

. Then, by Lemma 7.3

(2), we see that E

and D

must bound a K-parallelism in B. But this is

impossible, since D

contains the boundary components of a component of

K ∩ B. 

By the subclaim together with the arguments in Case 1, we see that the

genus one 1-bridge position X ∪

Y of K is K-isotopic to a genus one 1-

bridge position which is obtained from A ∪

B in a standard manner by

using the arc K ∩ (B ∩ X).

RUBINSTEIN–SCHARLEMANN GRAPHIC 143

Figure 8.8.

This completes the proof of Claim 1. 

Claim 2. If P ∩ Q consists of two components, then the genus one 1-bridge

position X ∪

Y of K is obtained from A ∪

B in a standard manner.

Proof. Let D

, A

, D

be the closures of the components of P −(P ∩Q) such

that D

and D

are disks, and A

is an annulus. Without loss of generality,

we may suppose that D

∪ D

is contained in X, and A

is contained in

Y . Since P is separating in S

, we see that both D

and D

are either

separating or non-separating in X.

Case 1. Both D

and D

are separating in X.

Let E be the disk in Q bounded by ∂D

. By changing subscript, if

necessary, we may suppose that ∂D

∩ E = ∅, i.e., D

is “outer” than D

Without loss of generality, we may suppose that E is contained in A. Then

we have the following case s.

Case 1.1. D

and E are K-parallel in X.

In this case, we can push D

along the parallelism out of X to make

P ∩ X = D

. Hence, we have the conclusion by Claim 1.

Case 1.2. D

and E are not K-parallel in X.

In this c ase , we ﬁrst claim that D

is K-incompressible in X. Assume

that D

is K-compressible in X. Then, by Lemma 8.6, the c omponent of

K − D

contained in X is isotopic rel ∂ to an arc in D

joining D

∩ K. We

denote by K

the image of K under this isotopy. Then K

is contained in A

and K

∩∂A(= K

∩D

) is an arc, and c`(K

∩IntA) is a trivial arc in A (see

Claim 1, Case 2.1). This implies that K is a trivial knot, a contradiction.

Hence D

is K-incompressible in X.

Let B

be the 3-ball in X bounded by D

∪ E. By Lemma 8.4, we see

that (B

, K ∩ B

) is a rational tangle. Ass ume that E is K-compressible

in A. Then by applying the last half of Lemma 7.4 to E in A, we see that

, K ∩B

) is a K-parallelism, c ontradicting the fact that D

and E are not

K-parallel in X. Hence E is K-incompressible in A. Then, by Lemma 7.3

(2), we see that E and c`(P − D

) bounds a K-parallelism in A. However

144 T. KOBAYASHI AND O. SAEKI

this is impossible, since the boundary components of a component of K ∩ A

is contained in c`(P − D

). This shows that Case 1.2 does not occur.

Case 2. Both D

and D

are non-separating in X.

In this case, we ﬁrst note that, since D

∩(X ∩K) (i = 1, 2) consists of two

points, the two points K∩Q are contained in a component of Q−(∂D

∪∂D

Then, let R be the closure of the component of X − (D

∪ D

) which does

not contain K ∩ Q. Without loss of generality, we may suppose that R is

contained in A. We have the following cases.

Case 2.1. Both D

and D

are K-incompressible in X.

In this case, we have the following subcases.

Case 2.1.1. D

and D

are not K-parallel in X.

Figure 8.9.

By Lemma 8.5, we see that (R, K ∩R) is a rational tangle. By Lemma 8.9,

we see that A

is parallel to the annulus R ∩ Q in Y , and, hence, P is K-

isotopic to ∂R. Now let a be the comp onent of K ∩ B that is contained in

X. Then, by Lemma 8.5 again, we see that the torus obtained from ∂R by

adding a tube along a is K-isotopic to Q. Hence we have seen that the genus

one 1-bridge position X ∪

Y of K is obtained from A ∪

B in a standard

manner by using the arc a.

Case 2.1.2. D

and D

are K-parallel in X, and there exists a K-incompress-

ible disk D

in X such that D

intersects K in two points, D

is non-

separating in X, D

∩ (D

∪ D

) = ∅, and D

is not K-parallel to D

(or

Figure 8.10.

RUBINSTEIN–SCHARLEMANN GRAPHIC 145

Let R

be the closure of the component of X − (R ∪ D

) which does not

contain K ∩ Q. By exchanging subscript, if necessary, we may suppose that

∩ R = D

. Let A

= R

∩ Q, and D

= A

∪ D

. We note that D

is a

disk properly embedded in B, which intersects K in two points. Since D

separates the boundary components of each component of K ∩ B in P , we

see that D

is K-incompressible. Moreover D

and D

are not K-parallel

in X, hence in B. Hence, by Lemma 7.3 (2), we see that D

and c`(P − D

)

(= D

∪ A

) are K-parallel in B. This shows that P is K-isotopic to ∂R

Then, by the argument of Case 2.1.1, with regarding R

as R we see that

the genus one 1-bridge position X ∪

Y of K is obtained from A ∪

B in a

standard manner.

Case 2.1.3. D

and D

are K-parallel in X, and there does not exist a

K-incompressible disk D

as in Case 2.1.2.

Figure 8.11.

In this case, it is easy to see that the argument of Case 2.1.1 works to

show that the genus one 1-bridge position X ∪

Y of K is obtained from

A ∪

B in a standard manner.

Case 2.2. Either D

or D

is K-compressible in X.

Without loss of generality, we may suppose that there is a K-compressing

disk D for D

∪ D

such that ∂D ⊂ D

By applying K-compression on D

, we obtain a compressing disk D for

∂X in X such that D∩K = ∅. By applying a slight isotopy, we may suppose

that D

∩ D = ∅. By Lemma 8.9, we see that A

and R ∩ Q are parallel in

Y . Hence, by K-isotopy, we may suppose that P = ∂R, (and A = R).

Then we have the following subcases.

Case 2.2.1. D is not contained in R.

146 T. KOBAYASHI AND O. SAEKI

Figure 8.12.

Since ∂D

and ∂D

are K-parallel, we see that D

is also K-compressible

in X. Then, by Lemma 8.6, we see that there is a disk ∆ in X such that

∆ ∩ D

= ∂∆ ∩ D

= γ an arc, and ∆ ∩ (K ∩ X) = c`(∂∆ − γ). Hence, we

can move K by an isotopy along ∆ such that ∆ ∩ (K ∩ X) is moved to γ

(hence, the comp onent of K ∩ B which inters ects Y is not changed by this

isotopy). Since each component of K ∩ B is a trivial arc, this shows that K

is a trivial knot, a contradiction.

Case 2.2.2. D is contained in R.

Figure 8.13.

Let R

= c`(X − R), and A

∗

= R

∩ ∂X. Then A

∗

is an annulus properly

embedded in B such that each component of ∂A

∗

is K-essential in ∂B,

and A

∗

intersects K in two points. By Lemma 7.3, we see that there is a

K-compressing disk D

for A

∗

in B.

Then we claim that D

is contained in X. In fact, assume that D

contained in Y . Since H

) = {0}, we see that ∂D

bounds a disk E in

∗

such that E ∩ K consists of two points, and D

∩ K = ∅. Let B

be the

3-ball in Y bounded by D

∪ E, and B

= c `(X − N(D)). By Lemma 8.3

(1), we see that K ∩ B

(= K ∩ X) is a trivial arc in B

. By Lemma 8.3 (3),

we see that K ∩ B

is a trivial arc in B

. Note that B

∪ B

is a 3-ball,

and B

∩ B

= E. This shows that K is a trivial knot, a contradiction.

Hence D

is contained in X. Since each component of ∂A

∗

separates the

boundary points of each component of K ∩ B in P , we see that D

∩ K 6= ∅,

RUBINSTEIN–SCHARLEMANN GRAPHIC 147

and, hence, D

∩ K consists of a point. By applying K-compression on A

∗

along D

, we obtain two disks D

, D

in B such that ∂D

= ∂D

, and

∂D

= ∂D

. Since D

(i = 1, 2) separates the boundary components of a

component of K ∩ B, D

is K-incompressible in B. Then, by Lemma 7.3

(2), D

∪ D

(i = 1, 2) bounds a rational tangle in B. Let B

300

be the 3-ball

obtained from X by cutting along D

. We regard D

, D

as contained in

∂B

300

, and D

, D

are properly embedded in B

300

. By Lemma 8.3 (2), we

see that (B

300

, K ∩B

300

) is a 2-string trivial tangle. Then we see that D

, D

are K-compressible, by the condition of Case 2.2. Hence, by the last half of

Lemma 7.4, we see that D

and D

resp.) are K-parallel. Let

a be the component of K ∩ B that is contained in X. Then, by the above

observations, we see that the ge nus one 1-bridge position X ∪

Y of K is

obtained from A ∪

B in a standard manner by using a.

This completes the proof of Claim 2. 

Completion of the proof of Proposition 8.8. Suppose that ](P ∩ Q) ≥ 3.

Note that the components of P ∩ Q are mutually K-parallel in P . Let D

, . . . , A

, D

be the closures of the components of P − (P ∩ Q) such that

, D

are disks and A

, . . . , A

are annuli that are located on P in this

order. Then we have the following cases.

Case 1. Either D

or D

is non-separating in the solid torus.

Without loss of generality, we may suppose that D

is contained in X,

and is non-separating in X. Then A

is contained in Y , and, by Lemma 8.9,

there is a homeomorphism from A

× I to Y such that A

corresponds to

× {1/2}. Let U be the closure of the component of Y − A

which does

not contain K ∩ Q.

Suppose that (IntU) ∩ P 6= ∅. Then we can push the component of

(IntU) ∩ P along the parallelism U to X to reduce ](P ∩ Q), yet still have

at least two components ∂A

Suppose that (IntU)∩ P = ∅. Then we can push A

along the parallelism

U to X to reduce ](P ∩ Q) by two.

In either case we have the conclusion by the assumption of the induction.

Case 2. Both D

and D

are separating in the solid torus.

If D

or D

is K-boundary parallel, then we can apply the assumption

of the induction by the argument as in Case 1. Hence we suppose that D

and D

are not K-boundary parallel. Let E

be the disk in Q bounded by

∂D

. Without loss of generality, we may suppose that E

⊂ E

, and D

contained in X. Then we have the following subcas es .

Case 2.1. D

is K-incompressible in X.

Let B

be the closure of the component of X − D

such that ∂B

∪ E

. Then, by Lemma 8.4, (B

, K ∩ B

) is a rational tangle. Suppose

148 T. KOBAYASHI AND O. SAEKI

(IntB

) ∩ P 6= ∅. We note that each component of c`((IntB

) ∩ P ) is an

annulus whose boundary components are parallel to ∂D

in Q − K. This

shows that each annulus is K-incompressible in B

. Hence, by Lemma 7.3

(3), we see that the closure of each component of (IntB

) ∩ P is boundary

parallel annulus. Hence, we can push them to Y to re duce ](P ∩ Q), and we

have the conclusion by the assumption of the induction. If (IntB

) ∩ P = ∅,

then by Lemma 7.3 (2), the last half of Lemma 7.4, and the assumption that

is not K-boundary parallel in X, we se e that c`(P − D

) is K-isotopic

to E

rel D

. Hence, by the argument in Claim 1, we see that the genus

one 1-bridge position X ∪

Y of K is obtained from A ∪

B in a standard

manner.

Case 2.2. D

is K-compressible in X.

We moreover have the following subcases .

Case 2.2.1. D

⊂ Y , and D

is K-incompressible in Y .

Let B

be the closure of the component of Y −D

such that ∂B

= D

∪E

In this case, s ince ∂D

⊂ IntE

, we see that (IntB

) ∩ P 6= ∅. Hence, by

the argument of Case 2.1 (for the case (IntB

) ∩ P 6= ∅), we can show that

the genus one 1-bridge position X ∪

Y of K is obtained from A ∪

B in a

standard manner.

Case 2.2.2. D

⊂ Y , and D

is K-compressible in Y .

In this case, by K-compressing D

, we obtain a K-compressing disk D

for ∂Y in Y such that ∂D

= ∂D

. Then, by Lemma 8.3 (3), we see that

K ∩ Y is rel ∂ isotopic to an arc α

in E

. And we also see that K ∩ X is

rel ∂ isotopic to an arc α

in E

such that α

∩ α

= ∂α

. Hence

K is a trivial knot, a contradiction.

Case 2.2.3. D

⊂ X.

Let T

and B

be the closures of the component of T − D

such that T

a solid torus and B

is a 3-ball. Without loss of generality, we may suppose

that B

-side of D

is contained in B. Since D

is K-compressible, we see

that K ∩ T

is rel ∂ isotopic in T

to an arc α in D

, by an isotopy that

does not move K − T

. Note that since D

⊂ X, c`(K − T

) is a component

of K ∩ B, hence, a trivial arc in B. This shows that K is a trivial knot, a

contradiction.

This completes the proof of Proposition 8.8. 

Proof of Theorem 8.2. Let A ∪

B be a Heegaard splitting which gives a 2-

bridge position of K, and X ∪

Y a Heegaard splitting which gives the given

genus one 1-bridge position of K. By Proposition 8.7, it is enough to assume

that X∪

Y is strongly K-irreducible for the proof of Theorem 8.2. Then, by

Proposition 7.5, and Corollary 6.22, we may suppose that P and Q intersect

RUBINSTEIN–SCHARLEMANN GRAPHIC 149

in non-empty collection of simple closed curves which are K-essential in both

P and Q. Then we have the conclusion by Proposition 8.8. 

Appendix A.

Let γ be a system of trivial arcs in a handlebody H, and p :

H → H the

two fold branched cover of H along γ.

Let F be a surface properly embedded in H, which is in general position

with respect to γ. Then, by using Z

-equivariant loop theorem [12], we see

that:

Lemma A.1. F is γ-incompressible if and only if

F (= p

−1

(F )) is incom-

pressible.

Moreover, by using Z

-equivariant cut and paste argument as in [9, Proof

of 10.3], we see that:

Lemma A.2. A γ-incompressible surface F is γ-boundary compressible if

and only if

F is boundary compressible.

By using Z

-Smith conjecture ([21], [16]) together with the Z

-equivariant

cut and paste argument and the irreducibility of H, we have:

Lemma A.3. A γ-incompressible surface F is γ-boundary parallel if and

only if

F is boundary parallel. In particular, if F is a disk intersecting γ in

one point, and ∂F bounds a disk D in ∂H such that D intersects γ in one

point, then F is γ-boundary parallel (in fact, F and D are γ-parallel).

Appendix B.

Let (B, β) be a 2-string trivial tangle, and (

β) the 2-fold branched cov-

ering space of B along β. Then

B is a solid torus,

β a system of two trivial

arcs in

B, and the covering translation τ is a π-rotation along

β (for details,

see [3, Chapter 12]).

Figure B-1.

150 T. KOBAYASHI AND O. SAEKI

We leave the proof of the next lemma to the reader.

Lemma B.0. Let F be an orientable incompressible surface properly em-

bedded in a solid torus. Then either:

1. F is a meridian disk,

2. F is a boundary parallel disk, or

3. F is a boundary parallel annulus.

Then, we s how:

Lemma B.1. Let D be a β-essential surface in B. Then D is a disk disjoint

from β, and D separates the compo nents of β.

Proof. Let

D be the lift of D in

B. By Lemmas A-1, A-3, we se e that

B is

an essential surface in the solid torus

B. By Lemma B-0, we see that

D is

a meridian disk. Suppose that

D ∩

β 6= ∅. Then we see that

D ∩ β consists

of a point, and, hence, D ∩ β consists of a point. However this implies that

∂D is β-inessential in ∂B. Hence, by Lemmas A-3, and B-0, we see that D

is β-boundary parallel, a c ontradiction. Since ∂D is β-essential in ∂B, we

see that ∂D separates the points ∂β in ∂B. This shows that D separates

the components of β. 

Lemma B.2. Let F be a β-incompressible surface in B.

Then either (0) F is β-essential, (1) F is a β-boundary parallel disk inter-

secting β in at most one point, (2) F is a β-boundary parallel disk intersect-

ing β in two points and F separates (B, β) into the parallelism and a rational

tangle, or (3) F is a β-boundary parallel annulus such that F ∩ β = ∅.

Proof. Let

F be the lift of F in

B. By Lemma A-1, we see that

F is one

of (1), (2), or (3) of Le mm a B-0. It is easy to see that (1) ((2) resp.) of

Lemma B-0 corresponds to the conclusion (0) ((1) resp.). Suppose that

is an incompressible annulus ((3) of Lemma B-0). Then it is easy to see

that we have conclusion (2) if

F ∩

β 6= ∅, and we have conclusion (3) if

F ∩

β = ∅. 

Lemma B.3. Let D be a β-compressible disk in B such that ∂D is β-

essential in ∂B, a nd D ∩ β consists of two points.

Then D separates (B, β) into two tangles (B

, β

), and (B

, β

), where

, β

) is a rational tangle such that there is a β-essential disk D

, β

) with D ∩ D

= ∅. Moreover if (B

, β

) happens to be a rational

tangle, then (B

, β

) is a β-boundary parallelism for D.

Proof. Let

D be the lift of D in

B. By Lemma A-1, we see that

D is a

compressible annulus in

B. Since ∂D is β-essential, we see that, by com-

pressing

D, we obtain two meridian disks, say D

and D

. Let B

, B

the closures of the components of

B − (D

∪ D

). Then B

, B

are 3-balls,

RUBINSTEIN–SCHARLEMANN GRAPHIC 151

and the closure of a component of

B −

D, say

T , is obtained from one of B

or B

, say B

, by adding a 1-handle, and hence

T is a solid torus, and there

is an equivariant compressing disk

for

D in

T such that τ (

) ∩

= ∅.

Hence

T /τ gives a rational tangle, and p(

) gives D

. Then the closure of

the other component of

B −

D, say E, is obtained from B

by removing a

regular neighborhood of an arc properly embedded in B

, and taking the

closure. This show s that E is homeomorphic to the exterior of a knot in

by a homeomorphism such that E ∩ ∂T is a regular neighborhood of

a meridian loop of the knot in ∂E. Suppose that E is a solid torus, i.e.,

(E, K ∩ E) is a rational tangle. Then the knot is a trivial knot. Since

E ∩ ∂T is a regular neighborhood of a meridian loop of the knot, we see

that E is a ∂-parallelism for

D, and by Lemma A-3, we see that (B

, β

) is

a β-boundary parallelism for D.

Appendix C.

Let α be a trivial arc in a solid torus T , and (

T , ˜α) the 2-fold branched

covering space of T along α. Then

T is a genus two handlebody, ˜α a 1-

string trivial arc in

T , and the covering translation is a π-rotation along

˜α.

Figure C-1.

Lemma C.1. Let D be an α-compressing disk for ∂T . Then D is either:

(1) a meridian disk of T with D∩α = ∅. In this case, we obtain, by cutting

(T, α) along D, a 1-string trivial tangle,

(2) a meridian disk of T with D ∩ α consists of one point, and we obtain

by cutting (T, α) along D, a 2-string trivial tangle, o r

(3) ∂-parallel disk in T with D∩α = ∅. In this case, D cobounds a 1-string

trivial tangle in (T, α).

152 T. KOBAYASHI AND O. SAEKI

Proof. Let

D be the lift of D in

T . Then we have either

D is a union of two

disks if D ∩ α = ∅ (Case 1), or

D is a disk if D ∩ α consists of one point

(Case 2).

In Case 1, we have either each component of

D is non-separating, or

separating, which correspond to the conclusions (1), (3) respectively. In Case

2, it is easy, by a homological argument, to see that

D is non-separating,

and this gives conclusion (2) 

Lemma C.2. Let D be a n α-essential disk in T such that D ∩ α consists

of tw o points. Then there exists an α-compressing disk D

for ∂T such

that D

∩ D = ∅, and D

∩ α consists of one point. Moreover, by cutting

(T, α) along D

, we obtain a 2-string trivial tangle (B, β) such that D is a

β-incompressible disk in (B, β) (hence D is β-boundary parallel).

Proof. Let

D be the lift of D in

T . By Lemma A-1, we see that

D is an

essential annulus in

T . Then it is easy to see that:

The annulus

D is obtained from a meridian disk ∆ by attaching a band.

(For a pro of of this, see, for example, [13, Lemma 3.2].)

Let

F = c`(∂

T − N(∂

D), where N (∂

D) is a regular neighborhood of ∂

in ∂

T . Note that

F is compressible in

T (in fact, slightly push oﬀ of ∆ gives

a compressing disk of

F ). Hence, by Z

-equivariant loop theorem [12], we

have an equivariant compressing disk(s) G for

F .

Claim 1. G consists of one disk, and, hence, G ∩ α consists of one p oint.

Proof. Ass ume that G consists of two disks D

, D

. Then we have the

following three cases.

Case 1. Each D

is separating in

T .

In this case D

and D

are parallel and the closures of

T − G are two

solid tori T

, T

, and a 3-ball B, which is a parallelism between D

, and

. Note that ˜α is contained in B, and this shows that

D is contained in

B, contradicting the incompressibility of

Figure C-2.

RUBINSTEIN–SCHARLEMANN GRAPHIC 153

Case 2. Each D

is non-separating, and G = D

∪ D

is non-separating in

T .

Since G ∩

D = ∅, this contradicts the incompressibility of

Figure C-3.

Case 3. Each D

is non-separating, and G = D

∪ D

is separating in

T .

In this case, we see that D

(i = 1, 2) intersects

β, a contradiction. 

Figure C-4.

Hence, G is a disk, and G ∩ α consists of one point. Then:

Claim 2. G is non-separating in

T .

Proof. Ass ume that G is separating. Then the closures of

T − G are solid

tori T

, T

with τ(T

) = T

. Moreover the ﬁxed point set of τ |

is an arc

in T

. But, by using Z

-equivariant loop theorem, it is easy to see that such

τ|

does not exist. 

By Claim 2, we see that we obtain a solid torus T

by cutting

T along G,

and this shows that we obtain a 2-string trivial tangle (B, β), and obviously

D is β-incompressible. 

Figure C-5.

154 T. KOBAYASHI AND O. SAEKI

Lemma C.3. Let D

, D

be mutually disjoint non α-parallel, α-essential

disks such that D

∩ α (i = 1, 2) consists of two points. Then there exists

an α-compressing disk D

for ∂T disjoint from D

∪ D

such that D

∩ α

consists of one point. Moreover each D

is non-separating in T , and, by

cutting (T, α) along D

, we obtain 2-string trivial tangle (B, β), and D

, D

are mutually non β-parallel, β-boundary parallel, β-incompressible disks in

(B, β).

Proof. Let

be the lift of D

T (i = 1, 2). By Lemma A-1, we see that

is an incompress ible annulus in

T . Let

F = c`(∂

T −N (∂

∪∂

, ∂

T )). The

argument in the proof of Lemma C-2 works in this case to s how that there

is an e quivariant compressing disk G for

F such that G is non-separating,

and G intersects ˜α in one point. Then let

be the solid torus obtained by

cutting

T along G, and G

, G

the copies of G in ∂

. By Lemma B-0 (3),

we see that there are annuli A

, A

in ∂

such that A

and

are parallel

(i = 1, 2). Since

are essential, we see that G

⊂ A

, G

⊂ A

. It is

easy to see that this gives the conclusion of Lemma C-3. 

Lemma C.4. Let D be an α-compressible disk such that ∂D is α-essential

in ∂T , and D ∩ α consists of two points. Then there is a disk ∆ in T such

that ∆∩D = ∂∆∩D = γ an arc, and ∆∩α = c`(∂∆−γ). Particularly, if D

is separating in T , then D separates (T, α) into (T

, α

), and (B

, α

) such

that α

is a trivial arc in a solid torus T

. In this case, if (B

, α

) happens

to be a rational tangle, then (B

, α

) is an α-boundary parallelism.

Proof. Let

D be the lift of D in

T . By Lemma A-1, we see that

D is a

compressible annulus in

T . It is easy to se e that there is a compressing disk

∆ for

D such that τ (

∆) =

∆, hence ˜α ⊂

∆. Then the projection of

∆ gives

∆. Since ∂D is β-essential, we see that, by compressing

D along

∆, we

obtain two meridian disks, say D

and D

, which are mutually parallel in

T .

Suppose that D is separating in T . Then

D is also separating in

T , and the

closures of the components of

B − (D

∪ D

) consist of B

and T

, where B

is a 3-ball, and T

is a solid torus. Then the closure of a component of

T −

say H, is obtained from T

by adding a 1-handle, and hence H is a genus two

handlebody. Now we consider the closure of the other component of

T −

say E. Then E is obtained from B

by removing a regular neighborhood of

an arc properly embedded in B

, and taking the closure. This shows that E

is homeomorphic to the exterior of a knot in S

by a homeomorphism such

that E ∩ ∂T is a regular neighborhood of a meridian loop of the knot in ∂E.

Suppose that E is a solid torus, i.e., (B

, α

) is a rational tangle. Then the

knot is a trivial knot. Since E ∩ ∂T is a regular neighborhood of a meridian

loop of the knot, we see that E is a ∂-parallelism for

D, and by Lemma A-3,

we see that D is α-boundary parallel. 

RUBINSTEIN–SCHARLEMANN GRAPHIC 155

References

[1] J.M. Boardman, Singularities of diﬀerentiable maps, Publ. Math. I. H. E. S., 33

(1967), 21-57.

[2] F. Bonahon and J.-P. Otal, Scindement s de Heegaard des espaces lenticulaires, Ann.

Sci. Ec. Norm. Sup., 16(4) (1983), 451-466.

[3] G. Burde and H. Zieschang, Knots, Walter de Gruyter, Berlin, New York, 1985.

[4] A. Casson and C. Gordon, Reducing Heegaard splittings, Topology and its Applica-

tions, 27 (1987), 275-283.

[5] J. Cerf, Sur les diﬀ´eomorphismes de la sph`ere de dimension trois (Γ

= 0), Lecture

Notes in Math., 53 (1968), Springer-Verlag, Berlin and New York.

[6] H. Doll, A generalized bridge number for links in 3-manifolds, Math. Ann., 294 (1992),

701-717.

[7] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate

Texts in Math., 14 (1973), Springer-Verlag, New York, Heidelberg, Berlin.

[8] W. Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology,

Math. Assoc. Amer., (1968), 39-98.

[9] J. Hempel, 3-manifolds, Princeton University Press, Princeton, N. J., 1976.

[10] M.W. Hirsch, Diﬀerential topology, Graduate Texts in Math., 33 (1976), Springer-

Verlag, New York, Heidelberg, Berlin.

[11] W. Jaco, Lectures on three-manifold topology, CBMS Regional Conf. Ser. in Math.,

43 (1980).

[12] P.K. Kim and J.L. Tollefson, Splitting the P.L. involutions of nonprime 3-manifolds,

Michigan Math. J., 27 (1980), 259-274.

[13] T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus

two, Osaka J. Math., 21 (1984), 437-455.

[14] H. Levine, Classifying immersions into R

over stable maps of 3-manifolds into R

Lect. Notes in Math., 1157 (1985), Springer-Verlag, Berlin, Heidelberg, New York,

Tokyo.

[15] J.N. Mather, Stability of C

∞

mappings. V: Transversality, Adv. Math., 4 (1970),

301-336.

[16] J.W. Morgan and H. Bass (Eds.), The Smith Conjecture, Pure and Applied Math.,

112 (1984), Academic Press.

[17] K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann., 289

(1991), 143-167.

[18] H. Rubinstein and M. Scharlemann, Comparing Heegaard splittings of non-Haken

3-manifolds, Top ology, 35 (1996), 1005-1026.

[19] , Comparing Heegaard splittings-the bounded case, preprint.

[20] H. Schubert, Knoten mit zwei Br¨ucken, Math. Zeit., 66 (1956), 133-170.

[21] F. Waldhausen,

Uber Involutionen der 3-Sph¨are, Topology, 8 (1969), 81-91.

156 T. KOBAYASHI AND O. SAEKI

[22] H. Whitney, On singularities of mappings of Euclidean spaces. I. Mappings of the

plane into the plane, Ann. of Math., 62 (1955), 374-410.

Received April 1, 1998 and revised June 24, 1999. The authors were supported by Grant-

in-Aid for Scientiﬁc Research, Ministry of Education, Science and Culture, Japan.

Department of Mathematics

Nara Women’s University

Kita-Uoya Nishimachi

Nara 630

Japan

E-mail address: tsuyoshi@cc.nara-wu.ac.jp

Department of Mathematics, Faculty of Science

Hiroshima University

Higashi-Hiroshima 739

Japan

E-mail address: saeki@top2.math.sci.hiroshima-u.ac.jp

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

GEOMETRIC REALIZATIONS OF FORDY–KULISH

NONLINEAR SCHR

ODINGER SYSTEMS

Joel Langer and Ron Perline

A method of Sym and Pohlmeyer, which produces geo-

metric realizations of many integrable systems, is applied to

the Fordy–Kulish generalized non-linear Schr¨odinger systems as-

sociated with Hermitian symmetric spaces. The resulting

geometric equations correspond to distinguished arclength-

parametrized curves evolving in a Lie algebra, generalizing

the localized induction model of vortex ﬁlament motion. A natu-

ral Frenet theory for such curves is formulated, and the general

correspondence between curve evolution and natural curvature

evolution is analyzed by means of a geometric recursion operator.

An appropriate specialization in the context of the symmet-

ric space SO(p + 2)/SO(p) × SO(2) yields evolution equations

for curves in R

p+1

and S

, with natural curvatures satisfying

a generalized mKdV system. This example is related to recent

constructions of Doliwa and Santini and illuminate s certain

features of the latter.

1. Introduction.

Shortly after it was discovered that the Korteweg-deVries equation could

be linearized via the spectral transform method [G-G-K-M], Shabat and

Zakharov [S-Z] showed that the method could also be applied to the (cubic)

non-linear Schr¨odinger equation,

(NLS) −iψ

= ψ

|ψ|

ψ.

Almost concurrently, Hasimoto [Has] discovered the connection between

NLS and the localized induction equation (LIE), an idealized model of the

evolution of the curved centerline of a thin vortex tube in a three-dimensional

ideal ﬂuid. (See [Bat] for a derivation, and [Ric] for the history of this equa-

tion, also known as the Betchov-Da Rios equation.) Denoting this evolving

centerline by γ(s, t) (where s is arclength along the curve and t is time), the

curve evolution in this model is described by

(LIE) γ

= γ

× γ

= κB,

where κ(s) is the curvature and B the binormal. Recall that along a space

curve, the Frenet frame {T, N, B} satisﬁes the equations T

= κN, N

157

158 JOEL LANGER AND RON PERLINE

−κT + τ B, B

= −τN. The LIE-NLS connection is this: If a curve γ with

curvature κ and torsion τ evolves according to LIE, then the associated

complex curvature function, ψ = κe

τ(u)du

, evolves according to NLS.

In view of the LIE-NLS correspondance, it is not surprising that LIE

manifests familiar integrability characteristics, but in geometric form: Soli-

ton solutions, a hierarchy of conserved Hamiltonians in involution, a recur-

sion operator generating the corresponding inﬁnite sequence of commuting

Hamiltonian vectorﬁelds – the localized induction hierarchy.

The Hamiltonian nature of LIE itself was introduced by Marsden and

Weinstein in [M-W]; the equation’s Poisson geometry was further elucidated

in [L-P1]; Yasui and Sasaki developed the structure of LIE in the setting of

hereditary operator, Hamiltonian pairs, and master symmetries [S-Y].

Other recent papers have addressed a variety of closely related geomet-

ric topics of a more concrete nature, including: Knotted soliton curves of

constant torsion [C-I]; planar, spherical, and constant torsion-preserving

curve evolution [L], [L-P3, L-P4]; integrable variational problems for curves

[Lan-S1, Lan-S2, Lan-S3]; pseudospherical surfaces and Weingarten sys-

tems [Per1, Per2], evolution of immersed Riemann surfaces in R

pre-

serving the Willmore integral [G-L]. It is by now clear that the localized

induction hierarchy is a rich source of examples and structure in the clas sical

diﬀerential geometry of curves and surfaces.

Here we consider natural generalizations of the LIE hierarchy in higher

dimensional spaces. Our starting point is the Fordy-Kulish [F-K] construc-

tion of a generalized nonlinear Schr¨odinger equation (gNLS) (with spectral

problem) associated to a Hermitian symmetric Lie algebra g. We apply

a technique due to Sym [Sym] and Pohlmeyer [Pohl], diﬀerentiation with

respect to the spectral parameter, which produces geometric realizations of

many integrable systems. By this route, we arrive at a generalized LIE hi-

erarchy for distinguished arclength-parametrized curves evolving in g, the

ﬁrst three terms of which are:

= γ

= −[γ

, γ

= −



sss

[γ

, [γ

, γ

]]



Here, [ , ] is the Lie bracket in g, and the subscript s denotes derivative by

a curve parameter which is unit speed with respect to the Cartan-Killing

form on g. In this setting, a direct generalization of Hasimoto’s result is

proved (Theorem 3), establishing the correspondence between the above

curve evolution equations and evolution of natural curvatures by equations

in the gNLS hierarchy; in particular, a curve evolving by the second order

ﬂow, gLIE, has curvatures satisfying gNLS.

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 159

Interestingly, as in the three-dimensional case , the odd-order ﬂows are

more amenable to geometrically meaningful reductions. In fact, by an ad

hoc reduction, in the class of symmetric spaces SO(p + 2)/SO(p) × SO(2),

we were able to fully realize our original goal; namely, we obtain geometric

evolution equations applicable to arbitrary smooth curves in E

and S

. For

the third order ﬂow, our equations take the form

= −



sss

kγ



= −

T +

)

Here, u

, . . . , u

n−1

are curvatures belonging to a natural frame T, U

, . . . ,

n−1

. We show (Theorem 5) that the corresponding natural curvature vec-

tor, u = (u

, . . . , u

n−1

), satisﬁes the vector modiﬁed Kortew eg-deVries equa-

tion:

(mKdV) u

= −



sss

|u|



Note that these simple equations for γ and u are given, ﬁnally, without

reference to a Lie algebra.

We now describe the contents of the paper. Section 2 is a review of

the Fordy-Kulish construction of generalized NLS equations. In Section 3,

we apply the Sym-Pohlmeyer geometrization procedure in the Fordy-Kulish

setting, and develop a natural Frenet theory for the resulting curves. In

Section 4, we introduce the geometric recursion operator for the generalized

LIE hierarchy, and derive key variation formulas. Section 5 treats the special

class of Hermitian symmetric spaces mentioned above, and the reduction

yielding curve evolutions in Euclidean spaces and spheres. We note that

our constructions in the latter case are related to recent work of Doliwa

and Santini [D-S]; in fact, our investigation developed out of an eﬀort to

better understand their equations. Since the completion of our paper, we

have learned from Chuu-Lian Terng of her own work (with K. Uhlenbeck)

on generalizations of LIE [T-U1, T-U2].

2. The Fordy-K ulish generalizations of NLS.

Following a standard framework in the theory of integrable systems, the

nonlinear soliton equations arise as c ompatibility conditions for an overde-

termined linear system

(LS) φ

= (λA + Q)φ, φ

= V φ.

This system involves two independent variables, s (“position”) and

t (“time”), and a scalar λ, the spectral parameter. The eigenfunction φ(s, t; λ)

has values in a Lie group G, while U(s, t; λ) = λA + Q(s, t) and V (s, t; λ)

have values in the Lie algebra g of G. Here Q is the potential, which

is meant to evolve isospectrally, hence the lack of λ-dependence. For the

160 JOEL LANGER AND RON PERLINE

Fordy-Kulish generalized NLS equations, g is taken to be the compact real

form of a complex semi-simple Lie algebra g

; in fact, g is required to be

a Hermitian symmetric Lie algebra, and A, Q, are speciﬁc to the struc-

ture of g. To recall brieﬂy some of the relevant features of this structure,

g has a decomposition as a vector space sum, g = k ⊕ m, of a com-

pact subalgebra k and complement m, satisfying the bracket conditions

[k, k] ⊂ k, [m, m] ⊂ k, and [k, m] ⊂ m. Also, k is associated with a sp ec ial

element A in h, a Cartan subalgebra of g; namely, k is the commutator

algebra of A: k = kernel(ad

) = {B ∈ g : [B, A] = 0}. Further, J = ad

satisﬁes J

= −Id, i.e., J is a complex structure on m . Such an element

A is ﬁxed to form (LS) and Q is required to be an m-potential, that is,

Q(s, t) ∈ m for all t. The set of m-potentials Q(s) is clearly a vector space;

we will refer to a tangent vectorﬁeld W as a m-ﬁeld. Some of the above will

be explained more explicitly, as required for specializations, below. (Also,

see [F-K], [Hel] for more details.)

Cross-diﬀerentiating the equations in (LS) gives the zero curvature con-

dition U

− V

+ [U, V ] = 0 or

(ZC1) Q

= V

− [λA + Q, V ].

With the aim of ﬁnding V in terms of Q, such that the compatibility

condition (ZC1) is satisﬁed, a polynomial ansatz is invoked: V =

j=0

(j)

(s, t)λ

n−j

. (Our indexing convention reverses the order of [F-K].)

The strategy here is to substitute this express ion for V into (ZC1), set the

coeﬃcients of powers of λ equal to zero, then solve recursively for the P

(j)

and, ﬁnally, obtain a nonlinear PDE for the m-ﬁeld Q (from the λ

term).

To carry this out requires the decomposition of g given above. Namely,

each P

(j)

is decomposed as P

(j)

= P

(j)

+ P

(j)

, with P

(j)

∈ m and P

(j)

∈ k.

Then, using the above bracket conditions and J

= −Id, one obtains the

equations:

(0)

= 0,

(j)

= −J(∂

(j−1)

− [Q, P

(j−1)

]), j = 1, . . . , n,

∂

(j)

= [Q, P

(j)

], j = 0, . . . , n,

= ∂

(n)

− [Q, P

(n)

Note the ﬁrst and third of these equations imply P

(0)

is necessarily a

constant in k – we will take the “obvious choice” P

(0)

= P

(0)

= A. Also,

a choice of “constant of integration” is made, at each stage, as we are ap-

parently required to compute an antiderivative to obtain P

(j)

. An essential

(and remarkable) feature of the recursion scheme is that the antiderivative

is explicit, and is polynomial in Q and its derivatives; here, we simply il-

lustrate this point with the important case n = 2. The required terms are

readily generated in the order: P

(1)

= Q, P

(1)

= 0, P

(2)

= −[A, Q

], P

(2)

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 161

[Q, [Q, A]] (the last antiderivative following from the Jacobi identity). The

induced evolution on the m-ﬁeld Q is then given by

(gNLS) Q

= −JQ

−

A ,

which we will refer to as the Fordy-Kulish generalized NLS equation. (In

[F-K], the NLS equations are displayed componentwise, rather than in vec-

tor notation.)

The above recursion scheme may be more compactly described by intro-

ducing a recursion operator

R =

which takes m-ﬁelds

X to m-ﬁelds:

X = −(∂

− ad

∂

−1

(Appropriate speciﬁcation of the antiderivative ∂

−1

depends on the context.)

Deﬁning

(j)

= JP

(j)

, we can now write the recursion scheme and nonlinear

equations as

(1)

= JQ,

(2)

X = Q

(j+1)

(

)

, j = 1, 2, 3, . . . ,

(j+1)

, j = 0, 1, 2, . . . .

The last of these equations deﬁnes the (j + 1)rst term in the Fordy-Kulish

NLS hierarchy, the above NLS equation being the third term.

It will be useful to have a concrete (and particularly simple) example at

hand for illustrating the main ideas in the next few sections; thus, we begin

our:

Running example.

For the “classical NLS”, we take g = su(2). We use the basis A =

−i

B =

−i

, C =

−i

, where σ

, σ

are the Pauli matrices

0 1

1 0

, σ

0 −i

i 0

, σ

1 0

0 −1

The bracket relations [A, B] = C, [B, C] = A, [C, A] = B, imply that

k = span(A) and m = span(B, C) deﬁne a Hermitian symmetric Lie algebra

structure on g = su(2). Writing Q = bB + cC, and plugging into the

generalized NLS equation yields:

= −J



|Q|



where |Q|

= (b

+ c

). In this case, we can identify m with the complex

numbers (Q with ψ = b + ic), and then J coincides with multiplication by

i. Using this identiﬁcation (and a time reversal), we obtain exactly (NLS),

given in the introduction.

162 JOEL LANGER AND RON PERLINE

We conclude this section by recording some useful general identities, to

be used in later s ections. First of all, we note that the above treatment of

the overdetermined linear system LS did not fully reﬂect the dependence of

φ on the three variables, s, t, and λ. Introducing W(s, t, λ) = φ

−1

, we

write down the augmented linear system

= Uφ, φ

= V φ, φ

= Wφ,

with corresponding compatibility conditions

(ZC2) U

− V

+ [U, V ] = V

− W

+ [V, W ] = W

− U

+ [W, U] = 0.

Secondly, the geometric objects related to LS will be expressed in terms

of conjugates of U, V , and W , for which we will use the following notational

shorthand: For B, C ∈ g, we write {B} = φ

−1

Bφ, and {B, C} = {[B, C]}.

The following lemma (whose proof involves straighforward diﬀerentiation)

states analogues of a standard principle of rigid body mechanics (with “time”

being s, t, or λ): absolute velocity = relative velocity + tran sferred velocity

([Arn], p. 128).

Proposition 1. For any g-ﬁeld B(s, t, λ),

i) {B}

= {B

} + {B, U},

ii) {B}

= {B

} + {B, V }, and

iii) {B}

= {B

} + {B, W }.

Finally, combining Proposition 1 with (ZC2) yields at once six simple

identities; three of these we will use, so we collect them in:

Proposition 2. For U, V , and W as above, we have

{V }

= {U

}, {W }

= {V

}, and {W }

= {U

3. Sym-Pohlmeyer curves.

Throughout this section, we “freeze time” in the deﬁnitions of the previous

section. In other words, we consider a time-independent potential Q(s), and

suppose φ = φ(s; λ) satisﬁes the linear system φ

= Uφ = (λA + Q)φ, for

each value of the “parameter” λ. Setting W (s, λ) = φ

−1

, we consider the

g-valued function

γ(s, λ) = {W } = φ

−1

By Proposition 2, we have γ

= {W }

= {U

} = {A}. If K is the Cartan-

Killing form on g, K(B, C) = tr(ad

), then by Ad-invariance of K,

K(γ

, γ

) = K(A, A) = constant. In fact, K(A, A) = tr((ad

)

) = −d,

where d = dim(m). Therefore, γ will be an arclength-parameterized curve

in g with respect to the rescaled form h, i = −

K. Henceforth, we refer to

any curve in the one parameter family γ(s, λ) as a Sym-Pohlmeyer curve,

and denote by T the unit tangent vector T = γ

= {A}.

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 163

To develop a Frenet theory for such curves, we ﬁrst use Propos ition 1

to obtain an expression for the curvature normal κN of a Sym-Pohlmeyer

curve,

κN = T

= {A}

= {A

} + {A, λA + Q} = {A, Q} = {

Q}.

We refer to the m-ﬁeld Q itself as a curvature coeﬃcients vector. Next,

we ﬁx a basis for g of the form A

= A, A

, . . . , A

, B

, . . . , B

, where the

span k and the B

span m. Since the Killing form K is deﬁnite (g is

compact), we can further specify the basis to be orthonormal with respect

to h, i. The curvature normal vector is now expressible as

κN =







j=1







j=1

where N

= {B

Next, we w rite the derivatives of the N

)

= {B

}

= {B

, U } = {B

, λA + Q},

as a linear combination of themselves and the vectors T

= {A

}, i = 1, . . . , c.

Finally, we write the derivatives

)

= {A

}

= {A

, Q}, i = 2, . . . , c.

At this point, we have a closed system of Frenet equations for the c + d =

dim(g) frame vectors T

, N

, involving only the curvature functions κ

, the

spectral parameter λ, and the structure constants of g.

Running example.

For Sym-Pohlmeyer curves in (su(2); −

∼

; h, i) with curva-

ture vector Q = bB + cC, the curvature normal is given by T

= κN =

−c{B} + b{C} = κ

+ κ

, with κ

= −c, κ

= b, N

= {B}, N

= {C}.

Our Frenet system is completed by the two equations (N

)

= −κ

T − λN

and (N

)

= −κ

T + λN

. For λ = 0, this is none other than the natural

Frenet system for curves in R

(see, e.g., [Bis]). For a general value of the

constant λ, such a system may be thought of as inertial, in that the rigid

body deﬁned by {T, N

, N

} (identifying s with time) has constant tangen-

tial component of angular velocity. The relationship to the classical Frenet

system can be written κ

+ iκ

= κe

iθ

, and N

+ iN

= (N + iB)e

iθ

, where

θ =

τ(u) + λdu; also, κ

= κ

+ κ

and τ = κ

−2

(κ

)

− κ

(κ

)

) − λ.

While κ, τ and {T, N, B} are uniquely deﬁned along a regular space curve γ

(with κ 6= 0), the curvatures κ

, κ

and frame vectors N

, N

are determined

(given λ) only up to multiplication by a complex unit, e

iα

– this freedom

corresponds to the choice of antiderivative in the above formulas. Aside from

this diﬀerence, the natural Frenet theory resulting from these deﬁnitions is

similar to the classical Fundamental Theorem for space curves. In partic-

ular, the set of unit speed curves Γ = {γ : R 7→ R

} can be parametrized

164 JOEL LANGER AND RON PERLINE

by the following data: initial position γ(0), initial frame T (0), N

(0), N

(0),

and shape κ

(s), κ

(s). For a given curve γ, this data is unique up to choice

of arclength parameter s, real parameter λ, and S

-parameter e

iθ

How does the Sym-Pohlmeyer construction ﬁt together with the above

parametrization of Γ? Since Q = κ

B − κ

C and λ are explicitly part

of the construction, it suﬃces to discuss the initial data γ(0), and T (0),

(0), N

(0). Writing T = Ad

−

A, N

= Ad

−

B, N

= Ad

−

C, we see

that the initial frame is determined by the initial condition on φ, via the

adjoint representation of SU (2). In fact, the two-to-one homomorphism

Ad : SU (2) 7→ SO(3) implies all initial frames are achieved (twice) as

φ(0) varies over SU (2). Next, allow the initial condition on φ to depend

on λ, and regard φ(0, λ) as an arbitrary curve in SU (2). Since φ

−1

de-

scribes the usual trivialization of the tangent bundle T SU (2), it follows that

γ(0, λ

) = φ

−1

(0,λ

)

is an arbitrary point in su(2)

∼

. In conclusion,

the Sym-Pohlmeyer curves are precisely the unit speed curves in R

, and

the correspondence between the Sym-Pohlmeyer construction and Γ is fully

described.

Remarks.

1) The above example is prototypical in some, but not all respects. In

general, Sym-Pohlmeyer curves constitute a very special subclass of the unit

speed curves in g

∼

c+d

– the latter cannot all be described by only d

curvatures. In fact, the tan gent indicatrix T (s) of a regular curve in R

c+d

can be any smooth curve in the unit sphere S

c+d−1

⊂ R

c+d

, whereas a Sym-

Pohlmeyer curve has tangent of the form T = {A} = Ad

−1

A. Now the

Ad-orbit of A can be identiﬁed with the Hermitian symmetric space G/K

(K having Lie algebra k). Thus, the tangent indicatrix of a Sym-Pohlmeyer

curve lies in G/K ⊂ S

c+d−1

⊂ g. In special cases , the above procedure may

produce a closed system with fewer than (c + d) frame vectors - this will be

true of our main construction of Section 5 - and the situation may resemble

the example more closely.

2) In the general case, it is reasonable to refer to k

, . . . , k

as natural

curvatures (though this term will have a more special meaning in Section

5). Note that the non-uniqueness of natural curvatures is described by

the group K (SU(1) = {e

iθ

} in the example). Spec iﬁcally, suppose φ

(λA + Q)φ, and c onsider the Sym-Pohlmeyer curve γ = φ

−1

. Now let

ϕ = φ

φ, where φ

∈ K is any constant element. Then γ = ϕ

−1

, and ϕ

satisﬁes the linear syste m ϕ

= (λA + Ad

Q)ϕ, as is easily checked. So the

“gauge transformation” φ 7→ ϕ = φ

φ leaves the curve γ unchanged, while

transforming the natural curvatures according to Q 7→ Ad

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 165

4. The recursion operator and variation formulas.

Next we “un-freeze” time, and apply the above constructions to time-de-

pendent potentials Q(s, t), obtaining two-parameter families of unit speed

curves γ(s, t, λ). The t-derivatives of these will b e called Sym-Pohlmeyer

(variation) ﬁelds. Note that Proposition 2 gives a formula for such vector-

ﬁelds:

= {W}

= {V

Also, V satisﬁes the zero curvature equation, V

= U

+ [U, V ] = Q

[λA + Q, V ]. Comparing k-components gives (V

)

= [Q, V

], i.e., the k-

component of V is determined by the m-c omponent of V , according to

= ∂

−1

[Q, V

]. Diﬀerentiation of this equation by λ shows that, similarly,

)

is determined by (V

)

: (V

)

= ∂

−1

[Q, (V

)

]. It is convenient to

introduce an operator K which takes m-ﬁelds to k-ﬁelds:

K(B

) = ∂

−1

[Q, B

Thus any Sym-Pohlmeyer ﬁeld is of the form Y = {K(B

) + B

}, for some

m-ﬁeld B

(modulo integration constant in K). Now the ab ove deﬁnitions

easily imply the following formula for the arclength derivative of such a

Sym-Pohlmeyer ﬁeld:

= {(B

)

} + {K(B

), Q} + λ{B

, A} = {C

where C

is an m-ﬁeld.

Remark. The result just obtained has the following (partial) interpretation

in the context of curve geometry (in a Riemannian m anifold). Suppose

γ(s, t) is any one-parameter family of arclength parametrized curves, and

let X be the vectorﬁeld X = ∂

γ. Then X

(the covariant derivative of X

with respect to the unit tangent T ) has no tangential component; in fact,

the condition for a vectorﬁeld X to be locally arclength preserving ([L-P2])

is hX

, T i = 0. Of course, a Sym-Pohlmeyer ﬁeld satisﬁes this condition:

hT, Y

i = h{A}, {C

}i = hA, C

i = 0, since k and m are orthogonal with

respect to the Killing form. In the special case c = 1, the Sym-Pohlmeyer

ﬁelds are exactly the locally arclength preserving vectorﬁelds, while for c > 1,

the Sym-Pohlmeyer vectorﬁelds form a strictly smaller class of vectorﬁelds.

For a Sym-Pohlmeyer curve γ in a Hermitian symmetric Lie algebra g,

we now deﬁne three ope rators on vectorﬁelds Y = {B} along γ.

(i) renormalization operator:

P({B}) = {K(B

) + B

} = {∂

−1

[Q, B

] + B

};

(ii) geometric recursion operator:

RY = −P([T, ∂

Y ]);

166 JOEL LANGER AND RON PERLINE

(iii) intertwining operator:

Z(Y ) = ad

(Ad

Y ).

The next lemma explains the nomenclature for Z:

Proposition 3. For Y a Sym-Pohlmeyer ﬁeld,

ZRY = (

R − λ)ZY.

Proof. Using the above computation of Y

, we have

ZRY = −ZP([T, Y

])

= −Z({A, (B

)

+ [K(B

), Q] + λ[B

, A]})

= −Z({J((B

)

+ [K(B

), Q]) + λB

})

= −J(J((B

)

− ad

∂

−1

) + λB

)

= (∂

− ad

∂

−1

− λJB

= −(∂

− ad

∂

−1

− JλB

= (

R − λ)JB

= (

R − λ)ZY.

Next, we consider a Sym-Pohlmeyer variation, γ(s, t, λ) = φ

−1

= {W},

and the corresponding Sym-Pohlmeyer ﬁeld (inﬁnitesimal variation) X =

= {V

Proposition 4. RX = {V +

A} , for some constant

A ∈ k.

Proof.

RX = −P



∂

∂s

∂

∂t



= −P([T, T

])

= −P([T, {A}

]) = −P([T, {A, V }])

= −P({A, [A, V ]}) = −P({J

V })

= P({V

}) = {K(V

) + V

} = {V +

A}.

This last step uses K(V

) = V

A, as observed above, with the arbitrary

“integration constant”

A ∈ k explicitly displayed here.

Theorem 1. Variation of curvatures formula:

The time variation of the “curvature coeﬃcients vector” Q induced by a

Sym-Pohlmeyer ﬁeld X = γ

= {V

} is given by

= ZR

X + [Q,

A].

In the “gauge term”, [Q,

A],

A ∈ k is a constant.

Proof. Combining Propositions 2 and 4, we have

(γ

) = ZR({V +

A}) = −ZP([T, {V +

])

= −ZP([T, {Q

} + {

A, Q}]) = −ZP({A, Q

+ [

A, Q]})

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 167

= −(ad

)

+ [

A, Q]) = Q

+ [

A, Q] .

The term [Q,

A] can be interpreted as follows. As explained in Remark

2 of the previous section, the non-uniqueness of natural curvatures for a

given curve corresponds to the se t of transformations Q 7→ Ad

Q, where

∈ K is a constant. For a curve evolving in time t, φ

should be treated

as a function of t as well (with initial value φ

t=0

= Id). Diﬀerentiation

of φ

with respect to t results in the term [(φ

)

t=0

, Q] in the inﬁnitesimal

variation of Ad

The appearance of the square of the recursion operator in this formula

suggests that between the curve γ and curvature coeﬃcients vector Q, there

is an intermediate object whose variation ought to be considered in this

context. The appropriate intermediate object is a Sym-Pohlmeyer frame

= {K

}, i = 1, . . . , c, , N

= {M

}, j = 1, . . . , d, as considered above.

Theorem 2. Variation of Frames formula:

If B ∈ g is constant, then the time variation of F = {B} induced by a

Sym-Pohlmeyer ﬁeld X = γ

is given by

= [F, RX] + {

A, B},

for some constant

A ∈ k; i.e., RX is essentially the “ Darboux vector” for

any Sym-Pohlmeyer frame along γ.

Proof. Using Propositions 1 and 4, we compute

= {B, V } = [{B}, {V }] = [F, RX] + {

A, B}.

We are now in a position to geometrize the Fordy-Kulish NLS hierarchy,

the ﬁrst few terms of which we list here for convenience:

(1)

= JQ,

(2)

= Q

(3)

= −JQ

−

(n+1)

(n)

For a Sym-Pohlmeyer curve γ with curvature vector Q and with λ = 0,

let vector ﬁelds X

be deﬁned along γ according to:

(0)

= {A} = T,

(1)

= {Q} = −[γ

, γ

(2)

= −



−

[Q, [Q, A]] + [A, Q

]



= −



sss

[γ

, [γ

, γ

]]



(n+1)

= RX

(n)

168 JOEL LANGER AND RON PERLINE

Now consider the curve evolution equation γ

= X

(n)

. By Theorem 1 and

Proposition 3, we can write the corresponding curvature evolution as

= ZR

(n)

+ [Q,

A] = ZR

(n+1)

(1)

+ [Q,

(n+1)

(1)

+ [Q,

A] =

(n+1)

(1)

+ [Q,

(n+2)

+ [Q,

A].

We summarize this result (suppressing the gauge term, [Q,

A]) as:

Theorem 3. Evolution of a Sym-Pohlmeyer curve (with λ = 0) by γ

(n)

corresponds to curvature evolution by Q

(n+2)

. In particular, the

generalized LIE,

(gLIE) γ

= −[γ

, γ

corresponds to the curvature evolution by gNLS, Q

= −JQ

−

A (the

analogue of Hasimoto’s result).

Running example.

The geometric recursion operator for curves in R

can be written: RX =

−P(T ×

∂

∂s

X). Here, × is the cross product in R

, and the reparameteriza-

tion operator P turns an arbitrary vectorﬁeld along γ, Y = fT + gU + hV ,

into a locally arclength preserving vectorﬁeld, PY =

(κ

g + κ

h)ds T +

gU + hV , by changing only the tangential component. Using the identiﬁ-

cation of m with the complex plane, the operator Z may be regarded as a

simple isomorphism betwee n normal vectorﬁelds Y = gN

+ hN

and com-

plex functions Z(Y ) = i(g + ih). On the other hand, Q has already been

identiﬁed with the complex function ψ = b + ic = κ

− iκ

. Using these def-

initions, the inﬁnitesimal variation of ψ induced by the vectorﬁeld X = γ

may be written as:

∂

∂t

ψ = ZR

X + irψ, r a real constant. This diﬀer-

ential formula for the Hasimoto transformation easily implies, e.g., that if

γ(s, t) evolves by LIE,

∂

∂t

γ = κB = −κ

+ κ

, then ψ (with λ = 0)

evolves according to NLS. [There is a minor diﬀerence b e tween the formulas

discussed here and those of [L-P1, L-P2, L-P3]. In those references, ψ

was the “complex curvature function” κ

+ iκ

= i(b + ic) mentioned in the

introduction. The diﬀerential formula was written in terms of this ψ (with a

minor diﬀerence in the deﬁnition of Z); if

∂

∂t

γ = κB, then NLS is satisﬁed

by the latter ψ as well (NLS being i-equivariant).]

5. Evolution of curves in R

p+1

and S

We have seen that the geometric realization of the Fordy-Kulish NLS hier-

archy is a sequence of evolution equations on the space of Sym-Pohlmeyer

curves in a (real compact) Lie algebra g. The Sym-Pohlmeyer curves have

curvature vectors Q which are m-valued. As stated in Sec tion 3, Sym-

Pohlmeyer curves in general form a proper subset of the set of all arc-length

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 169

parameterized curves in the Lie algebra g. In this section we describe a spe-

ciﬁc instance of our constructions which allows for a more complete anal-

ysis and full geometric interpretation. We will consider the Lie algebra

g = so(p + 2) with subalgebra k = so(p) ⊕ so(2) corresponding to the Her-

mitian symmetric space BDI. After describing the relevant structure and

commutation relations in appropriate detail, we give an explicit formula for

the the operator

restricted to a distinguished subspace of m-ﬁelds.We

then show that the Sym-Polhmeyer curves associated with appropriately re-

stricted curvature functions can be naturally considered as corresponding

to all curves in the Euclidean space R

p+1

, and the geometric realizations of

the terms in an associated mKdV hierarchy appear as quite natural evolu-

tion equations on curves. We explicitly compute the ﬁrst non-trivial term,

and show that it induces curvature evolution corresponding to a particu-

larly simple coupled mKdV system. This mKdV system is a rather special

reduction of a system which ﬁts into the general framework of [F-K], [A-F]

(though BDI is an exceptional case in the framework of [A-F]).

We consider so(p + 2) lying in gl(p + 2, R). We have gl(p + 2, R) commu-

tation relations [e

j,k

, e

l,m

] = δ

k,l

j,m

− δ

m,j

l,k

, 1 ≤ j, k, l, m ≤ p + 2, where

j,k

is the matrix with 1 in the jth row, kth column, zero otherwise. Setting

i,j

= e

i,j

− e

j,i

, we can express so(p+ 2) commutation relations in the form

j,k

, f

l,m

] = δ

j,m

k,l

+ δ

k,l

j,m

− δ

j,l

k,m

− δ

k,m

j,l

As it turns out, in addition to the natural notation for the so(p+2) basis,

i,j

}, 1 ≤ i < j ≤ p + 2, it will be c onvenient to have a notation adapted

to a particular decomposition of so(p + 2); thus we deﬁne

A = f

1,2

= f

1,j+2

, j = 1, . . . , p,

= f

k+2,2

, k = 1, . . . , p,

m,n

= f

m+2,n+2

, m, n = 1, . . . , p.

The so(p + 2) commutation relations now take the form:

[A, X

] = Y

, [A, Y

] = −X

, [A, K

m,n

] = 0,

, Y

] = δ

j,k

, X

] = [Y

, Y

] = −K

j,k

, K

m,n

] = δ

j,m

− δ

j,n

, K

m,n

] = δ

j,m

− δ

j,n

j,k

, K

l,m

] = δ

j,m

k,l

+ δ

k,l

j,m

− δ

j,l

k,m

− δ

k,m

j,l

Now consider the following subspaces of g = so(p + 2):

k = span{A} ⊕ span{K

m,n

= span{X

}, m

= span{Y

170 JOEL LANGER AND RON PERLINE

and

m = m

⊕ m

Part of the structure implicit in these deﬁnitions is summarized in

Proposition 5.

i) g = k ⊕ m, and k = so(2) ⊕ so(p);

ii) k is the commutator subalgebra of A in g;

iii) J = ad

satisﬁes J

= −I.

In particular, g admits a Hermitian symmetric Lie algebra structure (as

deﬁned in Section 2).

Remark. The proposition does not fully capture all the relevant structure

of g for the geometric considerations to follow. In this connection it should

be noted that the same Hermitian symmetric Lie algebra g = k ⊕ m arises

as a byproduct of the standard construction of so(p + 2) as compact real

form of so(p+ 2, C). However, a diﬀerent X, Y -decomposition of m appears,

which lacks the required properties; speciﬁcally, the X, Y -bracket relations

are not as simple as above.

Next, recall the recursion operator

R =

(ﬁrst introduced in Section

2) which takes an m-ﬁeld

X to an m-ﬁeld

X (and which depe nds on the

m-potential Q). Henceforth, we adopt the following:

Specialization. Q is an m

-valued potential (more brieﬂy, an m

-potential)

and

X is an m

-ﬁeld.

The fact that this specialization is preserved by

is an immediate con-

sequence of the following:

Proposition 6. Let Q =

(s)X

, and

X =

(s)X

. Then

X = −



∂

+ u

∂

−1

)



;

ii)

X = −



∂

(∂

∂

−1

))

∂

−1

∂

− u

∂

))



Proof. The proof is by straightforward c omputation; however, we include

it, since (i) depends on the nice bracket formula [X

, Y

] = δ

j,k

A, and (ii) in-

volves a noteworthy cancellation. Consider

X = (−∂

J +ad

∂

−1

The ﬁrst term, −∂

X, can immediately be written as −

∂

. The

second term can be rewritten as

∂

−1

X =

k,l,m

∂

−1

)[X

, [X

, Y

]]

k,l,m

∂

−1

)[X

, δ

l,m

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 171

= −

k,l,m

∂

−1

)δ

l,m

= −

∂

−1

Summing these two terms gives the desired formula (i).

To prove (ii), write

X = (−∂

J +ad

∂

−1

X, and note that the

ﬁrst term is −∂

X = −∂

(

( ∂

∂

−1

)) X

); this accounts

for the ﬁrst two terms of (ii). Next, using [X

, [X

, X

]] = [X

, −K

j,k

] =

i,k

− δ

i,j

, one computes

∂

−1

i,j,k

∂

−1

∂

+ u

∂

−1

)

(δ

i,k

− δ

i,j

)

l,j

∂

−1

∂

+ u

∂

−1

)

−

l,k

∂

−1

∂

+ u

∂

−1

)

l,k

∂

−1

∂

+ u

∂

−1

)

−u

∂

+ u

∂

−1

)

= −

l,k

∂

−1

∂

− u

∂

which is the last te rm of formula (ii).

We return now to the Fordy-Kulish NLS hierarchy. The second evolution

equation, Q

(2)

= Q

evidently preserves the space of m

-potentials.

In fact, writing Q =

(s)X

, the resulting evolution for the components

is given by (u

)

= (u

)

From the proposition, it now follows that the evolution Q

(4)

(2)

also preserves the space of m

-potentials, and the same is true for

all of the even evolution equations Q

(2n)

. In particular, we can apply

formula (ii) to the m

-ﬁeld

(2)

)

, obtaining

(4)

= −



∂

(∂

∂

−1

∂

))

172 JOEL LANGER AND RON PERLINE

+ u

∂

−1

∂

− u

∂

))



Substituting

for ∂

−1

∂

), and u

∂

− u

∂

for ∂

−1

∂

−

∂

), we obtain a simpler expression for

(4)

= −

∂

It follows that the evolution equation Q

(4)

in terms of the components

is a modiﬁed Korteweg-deVries system:

(mKdVS) (u

)

= −

∂

, i = 1, . . . , p.

Remark. So far, we have seen that the space of m

-ﬁelds (along m

potentials) is preserved by the operator

; as a consequence, the even

terms in the NLS hierarchy induce evolution equations on the space of m

potentials. It turns out that a corresponding result holds in the general

setting of Hermitian symmetric Lie algebras, with the X, Y-decomp osition of

m mentioned in the previous remark. For reasons already given, one cannot

generally expect such simple formulas and equations corresponding to those

just presented. But from our point of view, the most important diﬀerence

with the present case shows up in the construction of Sym-Pohlmeyer curves

from curvature data Q. Note the key role of the X, Y -bracket relations in

the following

Proposition 7. Consider an m

-potential Q =

with associated

Sym-Pohlmeyer curve γ (with λ = 0). The unit tangent vectorﬁeld T =

{A} and vectorﬁelds U

= {Y

}, i = 1, . . . , p , deﬁned along γ, satisfy the

following closed, linear system:

)

= −u

Proof. We have

= {A}

= {A

} + {[A, Q]}

} =

;

)

= {Y

}

= {(Y

)

} + {[Y

, Q]}

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 173













{[Y

, X

]}

= −

i,j

{A} = −u

Let Ψ be a (p + 1) × (p + 1) matrix which is the fundamental solution to

the matrix diﬀerential equation







0 u

· · · u

−u

0 · · · 0

−u

0 · · · 0







Ψ,

Ψ(s

) = I

p+1

= (p + 1) × (p + 1) identity matrix.

Then by the fundamental theorem of diﬀerential equations, we can express

the moving frame T (s), U

(s) as







T (s)

(s)







= Ψ







T (s

)







It follows that the Sym-Pohlmeyer curve γ ⊂ g actually lies in the

aﬃne space γ(s

) + R

p+1

, where R

p+1

is here identiﬁed as the span of

T (s

), U

), i = 1, . . . , p. Moreover, the equations given in the propo-

sition are the natural Frenet equations (see Running example in Section 3)

for a curve in R

p+1

with curvatures u

(s), i = 1, . . . , p and natural frame

T, U

, i = 1, . . . , p . Thus, we conclude that the Sym-Pohlmeyer curve γ

may be regarded as a general space curve in R

p+1

Since the natural Frenet equations are not so well-known, we take a mo-

ment to indicate some of their geometric signiﬁcance. To begin with, the

conclusion just reached dep e nds on an analogue of the classical Fundamental

Theorem of Curve Theory (for curves in n-dimensional Euclidean space). In

particular, every smooth curve γ in R

p+1

satisﬁes the above system for some

choice of curvature functions u

(s), i = 1, . . . , p , and is uniquely dete rmined,

up to congruence, by these functions. (The converse statement diﬀers a bit

from that of the classical theorem, in that the natural curvatures u

are

uniquely determined by a curve γ only after the frame T (s

), U

) has

been speciﬁed at some initial point γ(s

).) Note that T

= κN =

174 JOEL LANGER AND RON PERLINE

implies κ

)

, so one can recover the standard (ﬁrst) curvature from

natural curvatures.

What’s more important, the curvatures u

measure the sphericity of a

curve in R

p+1

. In particular, suppose that, for s ome j, u

= c

= constant 6=

0 . Then (γ(s) + (1/c

)

= T + (1/c

(−T ) = 0, so γ lies on a p-

dimensional sphere of radius 1/c

. Further, U

is the inward pointing unit

normal to the sphere along γ, and the remaining frame vectors determine a

natural Frenet system along the spherical curve γ in the sense of covariant

diﬀerentiation in the sphere. Namely,

∇

T =

i6=j

∇

= −u

T, i 6= j.

More generally, if u

= c

, u

= c

, . . . , u

= c

, then γ lies on a (p+1−l)-

dimensional sphere of radius (c

+ c

. . . c

)

−1/2

. In the exceptional case,

= u

= . . . u

= 0, γ(s) lies on a (p + 1 − l)-plane. This corresponds

to the case in classical Frenet theory in which the last l curvatures vanish –

here the order of the curvatures matters.

As an application of the above discussion, we are now in a position to give

a purely geometric version of our earlier variation formula, in the context of

Euclidean and spherical curves.

Theorem 4. Variation of curvatures formula:

Let M be Euclidean space of dimension d = (p + 1), or a round sphere of

dimension d = p. Denote by G the (constant) scalar curvature of M (so

G = 0 or G =

in the case of a sphere of radius r). Let γ

= X =

αT +

d−1

i=1

describe a variation of a curve in M through unit speed

curves, where U

, i = 1, . . . , (d − 1), is a natural frame along γ(s, t). Then

the induced variation of the associated natural curvatures u

is given by

)

= (∂

+ G)x

+ ∂

(αu

) +

d−1

l=1

∂

−1

∂

− u

∂

))

i,j

, i = 1, . . . , (d − 1).

In the gauge term

i,j

, the c

i,j

are constants with c

i,j

= −c

j,i

Proof. Combine Theorem 1 with Propositions 3 and 6, after taking account

of the above discussion.

We now merge the last few topics and consider the geometric evolution of

curves in R

p+1

and S

corresponding to the above mKdV system. According

to Theorem 3, the curve evolution γ

= X

(2)

corresponds to the curvature

evolution Q

(4)

. In the present context, a fully geometric interpretation

GEOMETRIC REALIZATIONS OF FORDY–KULISH SYSTEMS 175

of this result is possible; note that the following theorem (like the previous

one) is formulated entirely in terms of curve geometry – no Lie algebras!

Theorem 5. Motion of a curve γ(s, t) in R

p+1

by the geometric evolution

equation

= −



T + T



= −

T +

)

corresponds to curvature evolution by the mKdV system

)

= −



∂

|u|

∂



, i = 1, . . . , p.

Here, the functions u

are natural curvatures and k

= |u|

)

the

squared curvature of γ. In particular, if on e of the natural curvatures u

initially constant along γ, then this condition is preserved, and γ evolves on

a sphere.

Proof. The ﬁrst statement is easily obtained from the previous theorem, by

direct computation. Alternatively, in view of Theorem 3, it suﬃces to note

that in the present context,

(2)

= −



−

[Q, [Q, A]] + [A, Q

]



= −







−

i,j

, [X

, A]] +

)

[A, X

]







= −

(

)

A +

)

= −

)

T +

)

= −



T + T



It is also evident from the form of the coupled mKdV equations that the

condition u

= constant is preserved in time, so the last statement follows.

Remark. In the case k 6= 0, the ve ctorﬁeld X

(2)

is readily expressed in

terms of the standard Frenet frame:

(2)

= −



T + T



= −



T + κ

N + κτ B



Here, τ and B are, respectively, the second curvature and second normal (in

three dimensions, the torsion and binormal).

Expressed in this form, this vectorﬁeld appeared in [L-P1] as the “next”

term (above X

= kB) in the loc alized induction hierarchy, and the connec-

tion to the (complex) mKdV equation was discussed. The planar version of

176 JOEL LANGER AND RON PERLINE

this curve evolution and the connection to mKdV were considered in [G-P]

and in [L-P3]. In [L-P4], the authors showed that the even terms of the

localized induction hierarchy preserve curves lying on the sphere S

, ex-

pressing these vectorﬁelds in terms of the natural curvatures and frames. In

[D-S], Doliwa and Santini show that X =

T + κ

N + κτB describes an

evolution of curves on S

, and discuss corresponding curve evolution equa-

tions in spheres of arbitrary dimension. Our curve evolution equations (of

this section) apparently coincide w ith those of [D-S] (though our approach

is considerably diﬀerent). On the other hand, the corresponding curvature

evolutions of [D-S] are increasing complicated as dimension of the sphere in-

creases, and do not generally bear a close resemblance to the familiar (scalar)

mKdV equation; we attribute this to their use of standard Frenet systems,

rather than the natural Frenet systems we have employed here. Finally, our

general formalism suggests a new perspective on the main conclusions of

[D-S], regarding the characterization of integrable curve dynamics (as will

be discussed in a future paper).

In this work we have described a Lie-theoretic construction of generaliza-

tions of the localized induction hierarchy and, in a special case, have shown

how curvature evolution equations related to (but simpler in form than)

those of Doliwa and Santini may be extracted as a subhierarchy. These

geometric realizations of the Fordy-Kulish NLS systems have a structure

remarkably similar to the R

LIE equations studied previously. Given the

known relations between the localized induction hierarchy and classical geo-

metric constructions, it is not unreasonable to expect that the geom etric

realizations of the Fordy-Kulish NLS systems will have similar interesting

relations to geometry.

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dence, RI, 1996, 171-180.

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Phys., 35 (1994), 1732-1737.

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, J. London Math. Soc.,

30(2) (1984), 512-520.

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(1996), 605-618.

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for incompressible ﬂuids, Physica, 7D (1983), 305-323.

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dratic constraints, Comm. Math. Phys., 46 (1976), 207-221.

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562.

[S-Y] N. Sasaki and Y. Yasui, Diﬀerential geometry of the vortex ﬁlament equation,

J. Geom. Phys., 28(1-2) (1998), 195-207.

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one-dimensional self-modulation of waves in nonlinear media, Soviet Physics

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178 JOEL LANGER AND RON PERLINE

[Sym] A. Sym, Soliton surfaces and their applications, in ‘Geometrical aspects of

the Einstein equations and integrable systems’, Lecture Notes in Physics, 239

(1985), 154-231.

[T-U1] C.L. Terng and K. Uhlenbeck, Poisson actions and scattering theory for inte-

grable systems, preprint.

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Received September 2, 1998 and revised March 16, 1999.

Dept. of Mathematics

Case Western Reserve University

Cleveland OH 44106

E-mail address: jxl6@po.cwru.edu

Dept. of Mathematics and Comp. Sci.

Drexel University

Philadelphia PA 19104

E-mail address: rperline@mcs.drexel.edu

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

SEIBERG–WITTEN INVARIANTS FOR 3-MANIFOLDS

IN THE CASE b

= 0 OR 1

Yuhan Lim

In this note we give a detailed exposition of the Seiberg-

Witten invariants for cl osed oriented 3-manifolds paying par-

ticular attention to the case of b

= 0 and b

= 1. These are

extracted from the moduli space of solutions to the Seiberg-

Witten equations which depend on choices of a Riemannian

metric on the underlying manifold as well as certain pertur-

bation terms in the equations. In favourable circumstances

this moduli space is ﬁnite and naturally oriented and we may

form the algebraic sum of the points. Given any two sets

of choices of metric and perturbation which are connected

by a 1-parameter family, we analyse in detail the singular-

ities which may develop in the interpolati ng moduli space.

This leads then to an understanding of how the algebraic sum

changes. In the case b

= 0 a topological invariant can be

extracted with the addition of a suitabl e counter-term, which

we identify (this idea is attributed to Donaldson). In the

case b

= 1 a topological invariant is deﬁned which depends

only on cohomological information related to the perturba-

tion term. We prove a ‘wall-crossing’ formula which tells us

how the invariant changes with diﬀerent choices of this per-

turbation. Throughout we pay careful attenti on to genericity

statements and the issue of orientations and signs in all the

relations. The equivalence of t his invariant in the case of an

integral homology sphere with the Casson invariant is treated

in Lim, 1999 (see also works of Nicolescu, preprint). The

equivalence with Reidemeister Torsion in the case b

> 0 is a

result of Meng & Taubes, 1996. Some related material is in

Marcolli, 1996, Froyshov, 1996 and in the survey Donaldson,

1996. Taubes, 1990 contains the originating construction in

this article in the context of ﬂat SU (2)-connections.

1. The Seiberg-Witten Invariants.

We denote by Y an oriented 3-manifold. Let g be a Riemannian metric on

Y and P → Y a spin-c structure (see for example [LM]). Denote by S → Y

the associated positive spinor bundle, i.e., S = P ×

where % : spin

(3) →

End

) is the irreducible representation of the complex Cliﬀord algebra

179

180 YUHAN LIM

Cl(R

) with %(dy) = +1, where dy is the oriented volume form on Y . Let

ξ : spin

(3) = spin(3)×

{±1}

U(1) → U(1) be the map which takes the square

of the second factor. Then the bundle L = L

= P ×

C → Y is the

determinant line bundle of P ; it is clearly a U(1)-bundle. We remark that a

classical fact is w

(Y ) = 0 and this gives the existence of a spin-c structure

P → Y with c

(L) any given class in 2H

(Y ; Z) ⊂ H

(Y ; Z).

For a pair (A, Φ) consis ting of a U(1)-connection on L and a section of S

the π-perturbed Seiberg-Witten equations (SW

) read:

σ(Φ, Φ) + ω, D

A+α

Φ = 0.

Here F

denotes the curvature 2-form of A and D

the Dirac operator

coupled to A. The term σ(·, ·) is a certain symmetric bilinear form S →

(iR) (see Section 2 for details). The perturbation π consists of a pair

(α, ω) where α ∈ Ω

(iR) and ω ∈ Ω

(iR), dω = 0.

U(1) embeds into spin

(3) by the map ι(z) = [Id, z]. Given a smooth

map g : Y → U (1) ⊂ C, we have an automorphism of P given by the rule

p 7→ pι(g). This induces, by pulling back, the action (A, Φ) → g(A, Φ) =

(A + 2g

−1

dg, g

−1

Φ). If (A, Φ) is a Seiberg-Witten solution, then g(A, Φ)

is also a solution. Thus the set of Seiberg-Witten solutions is invariant

under the above automorphisms of S. The automorphism g called a gauge

transformation and the group of all gauge transformations is the gauge group

denoted G.

We will consider the set of L

-SW

-solutions modulo gauge equivalence

(details in Sec. 2). The set of such solutions will be denoted by Z

(P ; g)

or Z

π,g

(P ). When the underlying metric is understood we shall omit it and

simply write Z

(P ). Fixing a value of k ≥ 1, denote by P

the space of all

perturbations π of class C

Theorem 1. For π from an open dense subset of P

the irreducible part

∗

(P ; g) of Z

(P ; g) is a ﬁnite set of points and these are naturally oriented.

Let #Z

∗

(P ; g) denote the algebraic sum, assuming π as above. Then:

(i) if b

(Y ) > 1, #Z

∗

(P ; g) is independent of g and π

(ii) if b

(Y ) = 1, #Z

∗

(P ; g) depends only on the component of H

(Y ; R)\

(L)

} in which [

2π

ω] lies in, ω being the 2-form component of π

(iii) if b

(Y ) = 0, #Z

∗

(P ; g) is independent of π and g after the addition

of a counter-term ζ(π, g) which is a combination of the spectral invari-

ants of Atiyah-Patodi-Singer. #Z

∗

(P ; g) + ζ(π; g) takes values in Z if

(Y ; Z) = {0} and Z

8|H

(Y ;Z)|

if H

(Y ; Z) 6= {0}.

The exact expression for the counter-term ζ(π, g) is in Proposition 17.

For b

(Y ) = 1 the formula for the change in #Z

(P ; g) when we cross the

‘wall’ in H

(Y ; R) deﬁned by {c

(L)

} is given in Corollary 20.

SEIBERG-WITTEN INVARIANTS ... 181

Let spin

(Y ) denote the equivalence classes of spin-c structures on Y .

There is a well-deﬁned map spin

(Y ) → 2H

(Y ; Z) which sends a represen-

tative P to c

Corollary 2. Let Y be connected. The Seiberg-Witten equations deﬁne an

oriented diﬀeomorphism invariant τ of Y in the following form:

(i) if b

(Y ) > 0, τ : spin

(Y ) → Z

(ii) if b

(Y ) = 1, let spin

∗

(Y ) be the set of pairs ([P ], U) where U a com-

ponent of H

(Y ; R)\{c

)

}. Then τ : spin

∗

(Y ) → Z

(iii) if b

(Y ) = 0 and H

(Y ; Z) = {0} there is a unique spin-c structure

and so τ ∈ Z

(iv) if b

(Y ) = 0 and H

(Y ; Z) 6= {0}, then τ : spin

(Y ) → Z

8|H

(Y ;Z)|

In the subsequent se ctions we work up to a proof of Theorem 1. Sec-

tion 2 discusses the framework for deﬁning the moduli space and their ﬁrst

properties. Section 3 looks at generic properties. The details of the proof of

Theorem 1 are in Section 4.

Addendum. The reviewer has brought to the attention of the author of

an alternative exposition of some of the material in this article, in [Ch].

2. The Moduli Space.

Throughout this section Y denotes a closed oriented 3-manifold with Rie-

mannian metric g and P → Y a ﬁxed spin-c structure.

2.1. The Basic Set-up.

As in the usual gauge theory set-up we work with the following spaces.

(For more details see, for instance, [M].) Let C(P ) denote the space of

pairs (A, Φ) consisting of a L

connection A on L and Φ a L

- section of

S → Y . This forms a Hilbert manifold. Let G denote the space of L

gauge

transformations of S i.e., L

maps g : Y → S

⊂ C. This forms a Hilbert

Lie group. G acts on C(P ) by g(A, Φ) = (A + 2g

−1

dg, g

−1

Φ). This action is

smooth with Hausdoﬀ quotient B(P ).

A pair (A, Φ) is irreducible if Φ is not identically 0. Otherwise it is called

reducible. G acts freely on C

∗

(P ), the open set of irreducibles and its quotient

is denoted by B

∗

(P ). The projection map C

∗

(P ) → B

∗

(P ) forms a principle

G-bundle. At a reducible (A, 0), for which we simply write A, the stabilizer

of G is exactly those gauge transformations g for which dg = 0; thus the

stabilizer is identiﬁed with U (1) ⊂ C, the constant gauge transformations.

Let Ω

(iR) denotes the p-forms on Y of class L

, and Γ

(S) the sections

of S of class L

. Since C(P ) is an aﬃne space modelled on the vector space

Ω

(iR) × Γ

(S) the tangent space at any point is c anonically identiﬁed with

the vector space itself. On the other hand the tangent space to the identity

of G is identiﬁed with Ω

(iR).

182 YUHAN LIM

The derivative of the map g 7→ g(A, Φ) at the identity is given by γ 7→

(2dγ, −γΦ). The tangent bundle of C(P ) carries a natural Riemannian met-

ric which is the L

-inner product on Ω

(iR)×Γ

(S). This inner product is in-

variant under complex multiplication in Γ

(S). By taking the L

-orthogonal

to the image of the derivative of the gauge group action we obtain a slice

(A, Φ) + X

A,Φ

for the action at (A, Φ). X

A,Φ

is deﬁned as

{(a, φ) | 2d

∗

a = ihiΦ, φi} ⊂ Ω

(iR) × Γ

(S).

If Φ = 0 then X

A,Φ

reduces to ker d

∗

× Γ

(S). The stabilizer of A pre-

serves X

and acts by z(a, φ) = (a, z

−1

φ); therefore the stabilizer acts as

the opposite complex structure on Γ

(S). If Φ 6= 0 then a suﬃciently small

neighbourhood N of zero in X

A,Φ

models an open set for the gauge equiva-

lence class [(A, Φ)] in B(P ). If Φ = 0 then the same is true except that we

should take N/U(1) instead.

The symmetric bilinear form σ(Φ, Ψ) ∈ Λ

(iR) for us will be deﬁned as

the adjoint of Cliﬀord multiplication, that is deﬁned by the condition that

for all ω ∈ Λ

(iR),

hω · Ψ, Φi = hω, σ(Φ, Ψ)i.

The represe ntation c : Λ

(iR) → End

(S) given by Cliﬀord multiplication is

an isomorphism onto its image which is the trace-free Hermitian symmetric

endomorphisms of S. If we identify Λ

(iR) with its image under c then

σ(Φ, Ψ) =

(Φ ⊗ Ψ

∗

+ Ψ ⊗ Φ

∗

− hΦ, Ψi

Id).

In the formula expressions of the form v ⊗ w

∗

mean the endomorphism

v ⊗ w

∗

(u) = vhu, wi

. We remark that this formula assumes the convention

that if τ is unit length in Λ

(R) then c(iτ) is to be unit length in End

(S).

Fix a perturbation term π = (α, ω) of class C

, k ≥ 1 (we assume this

from now on). To set up the moduli space we deﬁne the SW

-section s =

π,g

: Ω

(iR) × C(P ) → Ω

(iR) ⊕ Γ

(S) by

s(η, A, Φ) =



∗



−

σ(Φ, Φ) − ω



+ 2dη, D

A+α

Φ − ηΦ



Since we will want to vary the perturbation term later, we introduce the the

Banach s pace Q

of C

-sections of Λ

(Y ) ⊗ iR and the Banach space Ω

closed C

-sections of Λ

(Y ) ⊗ iR. Then our perturbations π are from the

space P

which is Q

× Ω

From the deﬁnition of the SW

-section, it would seem that the zeros

might capture a much larger set than the SW

-solutions themselves; but as

the following Lemma shows this is only so in a minor way.

Lemma 3. Let s(η, A, Φ) = 0. If Φ 6= 0 then η = 0 and (A, Φ) is a SW

solution. If Φ = 0 then η = constant and (A, 0) is a SW

-solution.

SEIBERG-WITTEN INVARIANTS ... 183

Proof. We claim that the vector (∗(F

−

σ(Φ, Φ) − ω), D

A+α

Φ) is L

orthogonal to (2dη, −ηΦ). Before we show this we recall some useful identi-

ties: (i) if η ∈ Ω

(iR) then D

A+α

(ηΦ) = dη · Φ + ηD

A+α

Φ (ii) the Cliﬀord

action of a ∈ Ω

(iR) is equal to the action of − ∗ a (iii) if η ∈ Ω

(iR) then

hηΦ, Ψi = −hΦ, ηΨi. To prove the claim we compute:

h2dη, ∗(F

−

σ(Φ, Φ) − ω)i

− hηΦ, D

A+α

Φi

= h2η, d

∗

∗ (F

− ω)i

− h∗2dη,

σ(Φ, Φ)i

− hηΦ, D

A+α

Φi

= −

h∗dη · Φ, Φi

− hηΦ, D

A+α

Φi

hdη · Φ, Φi

− hηΦ, D

A+α

Φi

A+α

(ηΦ), Φi

−

hηD

A+α

Φ, Φ i

− hηΦ, D

A+α

Φi

A+α

(ηΦ), Φi

A+α

Φ, ηΦi

− hηΦ, D

A+α

Φi

= 0.

The last step follows from D

A+α

being self-adjoint. Therefore s(η, A, Φ) = 0

if and only if (A, Φ) is a SW

- solution and (2dη, −ηΦ) = 0. 

Thus we identify the space of Seiberg-Witten solutions with

−1

(0) ∩ {0} × C(P )

and the Seiberg-Witten moduli space Z

(P ) is the quotient by G of this,

where G acts only on the C(P ) factor. The local structure of the moduli

space near a solution (η, A, Φ) ∈ s

−1

(0) is determined by the elliptic complex

associated to the map s:

Ω

(iR)

A,Φ

−→ Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S)

η,A,Φ

−→ Ω

(iR) ⊕ Γ

(S)

where

A,Φ

(γ) = (0, 2dγ, −γΦ)

η,A,Φ

(ξ, a, φ) = (∗(da −

σ(Φ, φ)) + 2dξ, D

A+α

φ +

a · Φ − ξΦ − ηφ).

Since we will be interested only in the case where η = 0 we will in subsequent

notation omit it when that is understood. Thus we let H

A,Φ

, (i = 0, 1, 2)

denote the cohomology of the complex when η = 0.

Lemma 4. Let (A, Φ) be a SW

α,ω

-solution.

(i) If Φ 6= 0 then H

A,Φ

= 0, H

A,Φ

= H

A,Φ

(ii) If Φ = 0 then H

A,Φ

= H

(iR), H

A,Φ

= H

A,Φ

⊕H

A,Φ

, H

A,Φ

= H

(iR)⊕

A+α

Note. In this article H

(iR) always denotes the pure imaginary harmonic

forms of degree k and H

the kernel of D

184 YUHAN LIM

Proof. A direct computation shows that the L

-adjoint of δ

A,Φ

is given by

1∗

A,Φ

(b, ψ) = (2d

∗

b − ihiΦ, ψi, δ

A,Φ

(a, ψ)). The adjoint of δ

A,Φ

on the other

hand, is δ

0∗

A,Φ

(η, a, φ) = 2d

∗

a − ihiΦ, φi. There fore

A,Φ

= ker δ

0∗

A,Φ

∩ ker δ

A,Φ

= {(ξ, a, φ) | 2d

∗

a − ihiΦ, φi = 0, da −

σ(Φ, φ) = 0,

A+α

φ +

a · Φ = 0, dξ = 0, ξΦ = 0}

A,Φ

= ker δ

1∗

A,Φ

= {(b, ψ) | 2d

∗

b − ihiΦ, ψi = 0, db −

σ(Φ, ψ) = 0,

A+α

ψ +

b · Φ = 0}.

The remainder of the proof follows easily. 

The local structure of the moduli space may now be deduced by the

Kuranishi argument. Let (0, A, Φ) ∈ s

−1

(0). If Φ 6= 0, then a neighbourhood

of [(0, A, Φ)] in Z

(P ) is modelled on the zeros of a map (the obstruction

map) Ξ : H

A,Φ

→ H

A,Φ

. If Φ = 0 then the same is true except that Ξ is

-equivariant and we should take Ξ

−1

(0)/S

If (A, Φ) is a regular solution, i.e., H

A,Φ

= {0}, and Φ 6= 0 then Ξ

−1

(0) is

exactly one point. Thus a regular irreducible solution is isolated. Therefore

we have:

Proposition 5. If Z

∗

(P ) consists solely of gauge equivalence classes of reg-

ular solutions then Z

∗

(P ) is a discrete set, i.e., every point is isolated.

2.2. Compactness and Regularity.

Proposition 6. Fix π ∈ P

, k ≥ 1.

(i) If (A, Φ) is a SW

-solution then (A, Φ) is gauge equivalent to a SW

solution of class L

k+1

, p ≥ 2.

(ii) Let {(A

, Φ

)}

∞

i=1

be a sequence of SW

-solutions.

Then there is a subsequence {i

} ⊂ {i} and gauge transformations {g

} such

that {g

, Φ

)} converges in L

to a L

-solution. In particular, this

converges in C(P ), and therefore B(P ).

Proof. The proof is due to [KM]. We include it here for completeness. The

Bochner formula for the Dirac operator reads (see for instance [LM])

∗

A+α

Φ = ∇

∗

A+α

∇

A+α

Φ +

κΦ +

A+α

· Φ,

κ being scalar curvature. We also have Kato’s inequality

∆|Φ|

≤ h∇

A+α

∇

A+α

Φi.

SEIBERG-WITTEN INVARIANTS ... 185

If (A, Φ) is a SW

α,ω

-solution then D

A+α

Φ = 0 and

A+α

· Φ, Φi =

|σ(Φ, Φ)|

+ h(ω + dα) · Φ, Φi

|Φ|

+ h(ω + dα) · Φ, Φi.

Applying Kato’s inequality to the Bo chner formula we obtain

∆|Φ|

≤ −

κ|Φ|

−

|Φ|

+ |ω + dα|.|Φ|

At a maximum for Φ, ∆|Φ|

≥ 0. If this is non-zero we obtain

|Φ|

≤ max

(−2κ + 8|ω + dα|, 0).

Since ω and α are in C

, we obtain a uniform pointwise bound on the spinor

component of any SW

α,ω

-solution.

Let us prove (ii). Suppose that (A

, Φ

) is a given sequence of SW

α,ω

solutions. Choose a ﬁxed reference smooth connection A, and write A

A + a

. Then after a gauge transformation we may assume that da

= 0 and

the harmonic component of a

is uniformly bounded, since the component

group of maps Y → U(1) is H

(Y ; Z), and thus H

(Y ; iR)/H

(Y ; iZ) is

compact. Let ˆa

be the L

-component of a

which is L

-pependicular to the

harmonic forms. Since the harmonic forms are C

∞

, a

−ˆa

must lie in C

for

every k. The SW

α,ω

equations together with the uniform pointwise bound

on Φ

gives (by ellipticity) a uniform L

bound for the ˆa

. Applying this

to the equation for Φ

this gives again by ellipticity a uniform L

bound on

the Φ

. Circulating inductively we terminate with uniform L

k+1

bounds on

both ˆa

and Φ

, since α and ω are assumed to be in C

. The uniform bound

holds for all p ≥ 2. Therefore the sequence (a

, Φ

) is uniformly bounded in

k+1

. By Rellich’s theorem a subsequence of (a

, Φ

) converges in L

. For

p suﬃciently large, L

⊂ L

k+1

; thus the sequence converges in C(P ), since

the underlying topology in L

The proof of (i) follows from the preceding by applying it to the constant

sequence. 

2.3. Reducible Solutions.

When Φ = 0, the Seiberg-Witten equations reduce to a single equation

for the connection A: F

= ω. If ω = 0 then the reducible (up to gauge

equivalence) is identiﬁed with the moduli space of ﬂat U(1)-connections on

L. Thus a necessary condition is that c

(L)

= 0. If this is so, then by by

a well-known fact in diﬀerential geometry, the gauge equivalence classes of

ﬂat connections is completely determined by the holonomy representation

of π

(Y ) and is therefore topologically a product U (1) × · · · × U(1) where

the number of factors equals b

(Y ). In particular if b

(Y ) = 0 then the

186 YUHAN LIM

reducible is exactly one p oint. (Note: When b

(Y ) = 0, L admits only one

ﬂat connection, up to gauge.)

Lemma 7. The equation F

= ω has a solution if and only if the real coho-

mology classes [F

] = [ω] or equivalently [

2π

ω] = c

(L)

. If the latter holds,

then the space of equivalence classses of reducible solutions is topologically

U(1) × · · · × U(1) where the number of factors equals b

(Y ), and in the case

(Y ) = 0, a single point.

Proof. As explained above the c ondition [F

] = [ω] is necessary. For suﬃ-

ciency, let A

be such that [F

] = [ω]. Then we only need to solve for a in

= ω which is equivalent to da = ω − F

. Since ω − F

is exact such

an a can be found. Assuming solutions exist, let F denote the space of all

A’s such that F

= 0. Then F + a describes all the solutions to F

= ω.

Therefore the space of reducible solutions up to gauge are topologically the

same as in the case ω = 0. 

2.4. Orientation.

Suppose Z

∗

(P ) consists only of regular points. It is clear that Z

∗

(P )

is orientable. We want to produce a procedure for inducing a global ori-

entation. The fundamental elliptic complex can be combined into a single

operator L

η,A,Φ

: Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) → Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S),

η,A,Φ

= δ

η,A,Φ

+ δ

0∗

η,A,Φ

. A direct computation veriﬁes that this operator is

formally self-adjoint.

Let Λ = det Ind{L

η,A,Φ

}. Let g ∈ G and (a, φ) ∈ ker L

η,A,Φ

. Then

(a, g

−1

φ) ∈ ker L

η,g(A,Φ)

. Therefore the action of G lifts to an action on Λ.

Note that if Φ = 0 then the stabilizer U(1) maps the ﬁbre of Λ at A = (A, 0)

back to itself by the identity. Hence Λ decends to a line bundle

Λ over

Ω

(iR) × B(P ).

Proposition 8. The real line bundle

Λ is trivial.

Proof. We need to show that Λ posseses a G-equivariant trivialization. The

substitution of (1 − ε)Φ, 0 ≤ ε ≤ 1, for Φ and (1 − ε)η for η in the deﬁnition

of δ

η,A,Φ

and δ

0∗

η,A,Φ

deﬁnes a homotopy of L

η,A,Φ

to an operator L

η,A,Φ

given

by L

η,A,Φ

(ξ, a, φ) = (∗da + 2dξ, D

φ). This homotopy is G-equivariant. We

have ker L

η,A,Φ

= H

(iR) ⊕ H

= coker L

η,A,Φ

. This family has

a trivial determinant, and this proves Λ is G-equivariantly trivial. 

Notice that the homotopy given in the proof is the identity over {0} ×

Red

(P ). Thus over this set det Ind{L

η,A,Φ

} is the determinant of the index

of a constant family {L

dRham

} tensored with the complex family {D

Since a complex family is canonically oriented, we may ignore it. The kernel

and cokernel of L

dRham

are H

(iR) ⊕ H

(iR) and by identifying them with

each other we obtain a trivialization of det Ind{L

η,A,Φ

} over {0} × C

Red

(P ).

SEIBERG-WITTEN INVARIANTS ... 187

This orients det Ind{L

η,A,Φ

} over all of Ω

(iR) × C(P ). This is the natural

orientation of

Λ.

Consider the trivial real line bundle R over Ω

(iR) × B

∗

(P ). Then

Λ has

the property that over the open set O of Ω

(iR) × B

∗

(P ) deﬁned by the

condition that ker L

η,A,Φ

= 0, there is a canonical isomophism h :

Λ|

∼

−→

Proposition 9. Let

Λ have the natural orientation described above. As-

sume Z

∗

(P ) consists only of regular points. Then the following rule deﬁnes

an orientation ε : Z

∗

(P ) → {±1}. Let x ∈ Z

∗

(P ). Denote by o(x, R) the

canonical orientation of R|

and o(x,

Λ) the orientation induced by

Λ via

the isomorphism h above. Then

ε(x) =



1 if o(x, R) = o(x,

Λ)

−1 otherwise.

3. Generic Properties.

Let {g(t)}, t ∈ I

= (−ε, 1 + ε) be a 1-parameter family of metrics on Y . In

this section we examine the parameters (π, t) in P × I

for which the moduli

space Z

(P ; g(t)) consists solely of regular points.

In order to understand how the geometric structures and operators

changes with the 1-parameter family of metrics it will be useful to b e able to

work with a single reference underlying metric, spin-c structure and model

for the spinors. Fix an underlying metric which we take to be g, let P

the corresponding oriented orthonormal frame bundle and h an automor-

phism of T Y . h induces an automorphism h

∗

of P

, the component of

positively oriented frames of the frame bundle of Y . The image of P

under h describes the orthonormal frame bundle of another metric.

Conversely, the positive orthonormal frame bundle of any other metric can

be recovered on this way. Call the second metric g

h can be lifted to an isomorphism between P , the spin-c structure for g

and P

, the spin-c structure for g

. This, in turn, induces a ﬁbrewise isometry

h between the corresponding spinor bundles S and S

. By changing

h to

h where u is a smooth function on Y we can arrange it so that e

h gives

an isometry between Γ(S) and Γ(S

) with respect to their L

-norms.

Given the 1-parameter family g(t) the construction of e

h above can be

carried out smoothly in the parameter t, taking for instance g = g(0) to be

the reference metric. Therefore using these isomorphisms as identiﬁcations

we may assume (A, Φ) etc. for every t is deﬁned on a ﬁxed reference bundle.

The Dirac operator now depends also on t, and we denote this as D

g(t)

. It

is always self-adjoint with respect to the reference spinor bundle. Further

information regarding the relation between the Dirac operator for diﬀerent

metrics can be found in [B], [BG] and [H].

188 YUHAN LIM

3.1. Singular Locus of Dirac Operators.

In this section we discuss the singular locus for certain families of Dirac

operators, i.e., the parameters for which the Dirac operator is singular. This

will be crucial for us later.

Suppose b

(Y ) = 0. Since H

(Y ; iR) = 0, we have a bounded right

inverse d

−1

: Ω

→ Ω

(iR) for the op erator d. Let θ be a ﬁxed C

∞

ﬂat

connection on the determinant L. Then for any given ω, A = θ + d

−1

(ω)

solves F

= ω. Deﬁne {D(α, ω, t)} to be the family of Dirac operators

D : P × I

→ Fred

(Γ(S)), D(α, ω, t) = D

g(t)

θ+α+d

−1

(ω)

Here Fred

denotes the Banach space of Fredholm operators of index zero.

In the case b

(Y ) = 1, we shall also deﬁne a family as follows: This time

we keep the metric ﬁxed, so we drop it from the notation. Let A

be a ﬁxed

∞

connection on L and denote by ω

its curvature. Fix a choice of non-zero

∈ H

(iR) such that

4π

deﬁnes a generator for H

(Z). (The choice of

constants here is so that a

is the class of a gauge change 2g

−1

dg.) Then the

set {A

+ ta

| t ∈ [0, 1)}, parametrizes all the reducible SW

α,ω

solutions

up to gauge equivalence. Let E

, k ≥ 2, denote the exact forms in Ω

Then on E

we can as before deﬁne a bounded inverse d

−1

: E

→ Ω

(iR).

Given ω ∈ E

then A = A

+ ta

+ d

−1

(ω) solves F

= ω

. Thus the set

+ ta

+ d

−1

(ω

) | t ∈ [0, 1)} parameterizes up to gauge equivalence all

the reducible SW

α,ω+ω

-solutions, ω ∈ E

. Deﬁne the family {D(α, ω, t)} by

D : Q × E

× I

→ Fred

(Γ(S)), D(α, ω, t) = D

+ta

+α+d

−1

(ω)

Proposition 10. Let N be the subset of P ×I

consisting of all (α, ω, t) for

which D(α, ω, t) is singular. Similiarly deﬁne the subset K of Q × E

× I

for D(α, ω, t). Then N and K are nowhere dense closed subspaces.

Proof. We prove only the case for N . The other is done similiarly. Let B

be the unit L

-ball in Γ

(S). Let V → P × I

× B be the vector bundle

whose ﬁbre at (π, φ) is the real L

-orthogonal to φ in Γ

(S). By evaluating

D(α, ω, t) on φ ∈ B we obtain a section, call it D, of V . We claim this section

is transverse to the zero section. Let D(α

, ω

, t

)φ

= 0. Let ψ ∈ V

,ω

,φ

be L

-orthogonal to the derivative dD at (α

, ω

, t

, φ

). By varying φ

the tangent direction δφ we ﬁnd dD(δφ) = D(α

, ω

, t

)δφ; thus ψ must

also satisfy D(α

, ω

, t

)ψ = 0 (since D(α, ω, t) is self-adjoint). On the other

hand, by varying α

, dD(δα) = δα ·

and if ψ is L

-pependicular to this

then ψ = ifφ

for some real function f. The condition D(α

, ω

, t

)ψ = 0

then leads to df ·

φ = 0; but since φ

(x) 6= 0 on an open set it must be

that df = 0, and so f is a constant. Finally ψ being in V is necessarily

-orthogonal to φ

; thus f = 0. Hence ψ = 0 and transversality holds

and the zeros of D deﬁnes a smooth inﬁnite dimensional submanifold M of

P × I

× B. The projection map p : M → P × I

is proper since the kernel

SEIBERG-WITTEN INVARIANTS ... 189

of the Dirac operator is always ﬁnite dimensional. Applying the Sard-Smale

theorem we can concude that there exists an open dense set O in P ×I

with

the property that D(α, ω, t), (α, ω, t) ∈ O has nullity ≤ 1 over the reals. But

since this is a complex linear operator and self-adjoint, D(α, ω, t) must be

non-singular. The proposition now follows. 

Let π

= (α

, ω

), i = 0, 1, be given perturbations. Denote by {π(t)} =

{(α(t), ω(t))}, t ∈ I

the 1-parameter family of pe rturbations deﬁned by

π(t) = (1 − t)π

+ tπ

. For a ﬁxed value of π, D(π(t) + π, t) deﬁnes a 1-

parameter family of Dirac operators. We use the notation {D

(t)} for this 1-

parameter family. We call this family transverse (for the choice of π) if D

(0)

and D

(1) are non-singular and the family has transverse spectral ﬂow, as

t varies over [0, 1]. Transverse spectral also includes the condition that

multiple zero-eigenvalues do not occur as t varies. For the case b

(Y ) = 1,

ﬁx a value of α

. In a similar way we have a 1-parameter family {D

(t)}

obtained by considering D((α

, ω

) + π, t) for a ﬁxed π. Transversality is

deﬁned in the same way as before.

Proposition 11. Suppose {D

(t)} has the property that D

(t) is non-singu-

lar for t = 0, 1. Then there are arbitarily small π such that {D

(t)} is a

transverse family. A similiar statement holds for {D

(t)}.

Proof. We shall only prove the case of {D

(t)}. The other case is handled

by ess entially the same argument. Let x = (π

, t

) ∈ P × I, I = (0, 1).

Consider the map

G : Γ

(S) × P × I → Γ

(S), G(φ, π, t) = D(π, t)φ.

The diﬀerential of G at (0, π

, t

) is given by

dG(δφ, δω, δt) = D(π

, t

)δφ.

Then ker dG = H

⊕P ⊕R and coker dG = H

. Here H

denotes the kernel

of D(π(t

)+π

, t

) (acting on Γ

(S)). By the implicit function theorem there

is a neighbourhood V of (0, x) ∈ H

× P × I and a unique smooth map

f : V → H

⊥

such that for (φ, π, t) ∈ V ,

(I − Π)G(φ + f(φ, π, t), π, t) = 0, Π = L

-projection onto H

.(1)

Note that the linear extension of f in the φ variable continues to satisfy (1)

so we may take V to be of the form H

× W, W a neighbourhood of x.

Because of (1) the injective/surjective properties of D(π, t), (π, t) ∈ W , are

completely determined by the ﬁnite-dimensional operator ﬁnite dimensional

operator H

→ H

T (π, t)φ = ΠG(φ + f(φ, π, t), π, t).

We claim T (π, t) is self-adjoint with respect to the (real C-invariant) L

inner product on H

. We introduce the notation h·, ·i

, ψ, φ ∈ H

, to

190 YUHAN LIM

denote the L

inner product on H

and h·, ·i

Γ(S)

the L

-inner product on

Γ(S). We compute:

hT (π, t)φ, ψi

= hG(ψ + f(ψ, π, t), π, t), φi

Γ(S)

= hG(ψ + f(ψ, π, t), π, t), φ + f(φ, π, t)i

Γ(S)

since G(ψ + f(ψ, π, t), π, t) ∈ H

, f(φ, π, t) ∈ H

⊥

= hψ + f(ψ, π, t), G(φ + f(φ, π, t), π, t)i

Γ(S)

by self-adjointness of D(π, t)

= hψ, T (π, t)φi

This proves the claim.

Since T (π, t) is complex linear and self-adjoint with respect to the real

inner product, it is Hermitian with resp ec t to the natural complex extension

of the L

-inner product on H

. Let Herm(H

) denote the (real) vector

space of Hermitian transformations on H

. The determinant function det :

Herm(H

) → R ⊂ C and det

−1

(0) is a closed subvariety of codimension 1

in Herm(H

). Introduce the notation

(k)

= {l ∈ Herm(H

) | dim ker(l) ≥ k}.

Lemma 12. Suppose dim

) > 0. The derivative (dT )

of T re-

stricted to Q has non-trivial image in Herm(H

). If dim

≥ 2 then this

image is of dimension ≥ 2.

First let us show that the image of (dT )

is non-trivial. The derivative

at (π

, t

) is computed to be

(δ α, δt)φ = Π(δα + d

−1

δω) ·

φ, π = (α, ω).

Suppose that hdT

(δα)φ, φi

= 0 for all δα ∈ Q. Since

hδα ·

φ, φi

hδα, σ

(φ, φ)i.

This implies σ

(φ(y), φ(y)) = 0 for all y ∈ Y and thus φ = 0. Thus the

image of dT

is non-trivial.

Assume now that dim

) ≥ 2. Let δα be such that dT

(δα) 6= 0.

Let φ

, . . . , φ

be an complex orthonormal basis for Herm(H

) such that

(δα) is diagonal with respect to this basis. Thus hδα · φ

, φ

= 0

for i 6= j. Since the φ

are harmonic spinors, unique c ontinuation implies

that there is a open set in Y on which φ

6= φ

, i 6= j on this open set, in

particular say at the point y ∈ Y . We can ﬁnd a δα

with support in an

arbitarily sm all neighbourhood of y such that

hδα

· φ

, φ

6= 0, i 6= j.

Thus dT

(δα

) is independent of dT

(δα). This shows that the image

is at least 2-dimensional. This proves the Lemma.

SEIBERG-WITTEN INVARIANTS ... 191

Consider for each (α, ω) ∈ W,

α,ω

(t) = T (α, ω, t), |t − t

| < ε

where ε > 0 is chosen so that (α, ω, t) ∈ W . Clearly τ

α,ω

deﬁnes a path in

Herm(H

By an open cover argument the following exists: (1) a ﬁnite set {t

, . . . , t

}

⊂ [0, 1] together with open neighbourhoods V

of t

in R such that {V

}

covers [0, 1] (2) an open neighbourhood W

⊂ W of 0 ∈ P (3) maps

: W

× I

→ Herm(H

) as in the preceding which preserves the in-

jectivity/surjectivity properties of D(α, ω, t), (α, ω, t) ∈ W

× I

. We denote

the corresponding paths by τ

α,ω

(t).

Let N = max

{dim

)}. Suppose N > 1. Let j be such that

dim(H

) = N. Note that N

(N)

= {0} ⊂ Herm(H

). Thus any non-

zero element in Herm(H

) lies in N

(k)

, 0 ≤ k < N . Then by the above

Lemma we can ﬁnd a suﬃciently small perturbation (α, 0) ∈ W

so that

α,0

(t) ∈ N

)

, 0 ≤ k

< N , t ∈ I

and for i 6= j, τ

α,0

(t) ∈ Herm(H

)

for all t ∈ I

. Thus we establish that there is an arbitarily small α so that

α,0

(t) ∈ N

(k)

with 0 ≤ k < N for every i. Repeating the above con-

struction over but with W

taken to be an open neighbourhood of (α, 0)

instead, we inductively prove that we can ﬁnd an arbitarily small α

so that

we have N = 1. The perturbation argument in this case makes each path

transverse to N

(1)

= {0} ⊂ Herm(H

)

∼

Let us show that {D

(t)} is a transverse family. Suppose at s, τ

(s) =

0. Let φ ∈ H

be unit length. Let λ(t) be the 1-parameter family of

eigenvalues satisfying D

(t)φ = λ(t)φ for t close to s. Thus we have

(t)φ, φi

= λ(t)hφ, φi

Diﬀerentiating this equation with respect to t and evaluating at t = s gives

hdT

(0, 0, 1)φ, φi

= λ

(s).

The left hand term is simply the velocity of τ

at t = s and transversality

means this is non-zero. Thus λ

(s) 6= 0 and we have transverse spectral ﬂow

at t = s. 

3.2. The Parameterized Moduli Space.

As before we assume the 1-parameter families {g(t)}, t ∈ I

. Deﬁne the

parametrized Seiberg-Witten section to be the map

˜s : Ω

(iR) × C(P ) × P × I

→ Ω

(iR) ⊕ Γ

(S),

˜s(η, A, Φ, π, t) = s

π,g(t)

(η, A, Φ).

Then the parameterized moduli space is Z(P ) = ˜s

−1

(0)/G (with G acting

only on the C(P ) factor). There is the projection map p : Z(P ) → P × I

and clearly p

−1

(π, t) = Z

(P ; g(t)).

192 YUHAN LIM

Proposition 13. The irreducible part Z

∗

(P ) of the parameterized moduli

space is smooth Hilbert space manifold and the projection map p|

∗

(P )

∗

(P ) → P × I

is a smooth Fredhom map of index zero.

If b

(Y ) = 0 we determined in Prop. 10 that p

−1

(π) contains no reducibles

if we are oﬀ the set N ⊂ P × I

. If b

(Y ) > 0 we observed in Lemma 7 that

there are no reducibles in p

−1

(α, ω) if and only if [

ω] is not an integral

class in cohomology. Thus:

Lemma 14. Let q : P

× I

→ H

(Y ; R), q(α, ω, t) = [

2π

ω] and set W =

−1

)). Then p

−1

(π, t) has reducibles if and only if (π, t) ∈ W. W is

a closed nowhere dense subset of of codimension equal to b

(Y ).

We remark that in the case b

(Y ) > 0 the condition of regularity (i.e.,

A,Φ

= {0}) for reducible solutions can never be satisﬁed. This is because

when Φ = 0, H

A,Φ

reduces to H

(iR)⊕H

. So regularity implies the absence

of reducibles in this case.

Corollary 15. p|

∗

(P )

: Z

∗

(P ) → P × I

is proper over P × I

\W where

(i) W = N if b

(Y ) = 0 (ii) W = W if b

(Y ) > 0. Therefore for an open

dense set O ⊂ P × I

, p

−1

(z), z ∈ O, is a ﬁnite set of regular points.

Proof. The properness assertion is the content of Proposition 6 and the

fact that a regular reducible point in the case b

(Y ) = 0 is necessarily

isolated, by the Kuranishi local model. The Sard-Smale theorem then gives

the ‘open dense set’ statement since regularity of an irreducible solution

(A, Φ) is equivalent to the derivative d˜s at (0, A, Φ) being surjective. (Note:

without the properness assertion we can only conclude regularity on a Baire

set.) 

Proof of Proposition 13. We have to show that the derivative d˜s is surjective

at every point (0, A

, Φ

, π

, t

) ∈ ˜s

−1

(0) for which Φ

6= 0. Let (b, ψ) lie in

the cokernel of ds

0,A

,Φ

, i.e., (b, ψ) ∈ H

A,Φ

, thus

(i) db =

σ(Φ, ψ), (ii) D

ψ +

· Φ = 0, (iii) 2d

∗

b = ihiΦ, ψi.(2)

Suppose (b, ψ) is L

-orthogonal to the image of d˜s. The Proposition is proven

as soon as we can show (b, ψ) = 0. If δω ∈ Ω

, then d˜s(δω) = (∗δω, 0). Thus

b must be L

orthogonal to all the co-closed forms; this implies that b must

be closed. Then from (i) we obtain the condition σ

(Φ

, ψ) = 0. Working

at a point, the kernel of the transformation v 7→ σ

(w, v) is of dimension 1

and it is easy to check that σ

(w, iw) = 0. Therefore ψ = ifΦ

for some

real valued function f. Putting this into (ii) of (2) we obtain

0 = D

g(t

)

(ifΦ) +

= (idf +

) · Φ (since D

g(t

)

= 0).

SEIBERG-WITTEN INVARIANTS ... 193

Hence we obtain the pointwise condition idf +

= 0 on the open dense set

O where Φ

6= 0. By continuity it holds on all of Y . Substituting into (iii)

of (2) we get the equation

4∆f = −|Φ|

Taking the pro duct with f and integrating we obtain:

4|df|

+ |Φ|

|f|

= 0.

Thus f = 0 on O and therefore Y and we ﬁnally obtain (b, ψ) = 0. Finally

the index zero assertion follows directly from Lemma 4. 

4. Proof of Theorem 1.

Y is assumed to be a closed oriented 3-manifold with Riemannian metric

g and spin-c structure P → Y . According to Corollary 15 applied to the

constant family {g(t) = g}, we may choose a perturbation π from an open

dense set in P

(k ≥ 3) such that Z

π,g

(P ) consists of a ﬁnite set of regular

points, i.e., the cohomology H

A,φ

is trivial at these points. For this π, if

(Y ) = 0 there is a unique isolated reducible (up to gauge equivalence) and

if b

(Y ) > 0 there are no reducibles. Z

∗

π,g

(P ) is then naturally oriented by

our conventions (Proposition 9) and we can form the algebraic sum

π,g

(P ).

To prove the claimed invariance properties of #Z

π,g

(P ), let g

, g

be two

metrics on Y and let π

be two perturbations which satisfy the above with

respect to g

. We want to relate #Z

(P ) and #Z

(P ). Consider

the 1-parameter family of metrics {g(t)} = {(1 − t)g

+ tg

} deﬁned for

t ∈ I

= (−ε, 1 + ε). Thus as in Section 3 we have a parameterized moduli

space Z(P ) and projection map p : Z(P ) → P

× I

We consider a smooth path σ : [0, 1] → P

× I

, σ(0) = (π

, g

), σ(1) =

(π

, g

). We introduce the notation Z

(P ) for the σ-parametrized mo duli

space {(x, t) | x ∈ p

−1

(σ(t))}. If a portion of σ misses the ‘singular’ sets N

or W of Sec.3.1 and is transverse p, then that portion of Z

∗

(P ) consists

purely of regular points and therefore is a smooth arc. This is oriented in

the following way. The local deformation theory of Z

(P ) is described by

an elliptic complex of the form

Ω

(iR) → Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) ⊕ R → Ω

(iR) ⊕ Γ

(S).

Therefore the orientation is determined by looking at the ‘wrapped up’ op-

erator

η,A,Φ,t

: Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) ⊕ R → Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S).

194 YUHAN LIM

The orientation of the regular irreducible points of Z

(P ) is determined by

an orientation of the determinant of the index of the family {L

η,A,Φ,t

}. We

have the short exact sequence of 2-step complexes:

0 −−−−→ Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S)

η,A,Φ

−−−−→ Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) −−−−→ 0

0 −−−−→ Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) ⊕ R

η,A,Φ,t

−−−−−→ Ω

(iR) ⊕ Ω

(iR) ⊕ Γ

(S) −−−−→ 0

0 −−−−→ R −−−−→ 0 −−−−→ 0.

This gives rise to a canonical isomorphism

h : ker L

η,A,Φ,t

⊕ coker L

η,A,Φ

→ ker L

η,A,Φ

⊕ R ⊕ coker L

η,A,Φ,t

.(3)

An orientation for det Ind{L

η,A,Φ

} deﬁnes an orientation for det Ind{L

η,A,Φ,t

}

according to this rule: Choose an orientation for coker L

η,A,Φ

. Then an ori-

entation of ker L

η,A,Φ

is determined, since det Ind{L

η,A,Φ

} is oriented. Now

given an orientation of coker L

η,A,Φ,t

, then ker L

η,A,Φ,t

is oriented so that h is

an orientation-preserving isomorphism, where the domain and range spaces

are given the product orientation in the order written in (3). With this ori-

entation convention, if Z

consists entirely of regular irreducible points and

if compact then its boundary is precisely Z

σ(1)

(P ) − Z

σ(0)

(P ), as oriented

spaces.

The proof of Theorem 1 in the case b

(Y ) > 1 can now be eas ily estab-

lished. By Lemma 14 σ may be chosen to be disjoint from the subset of W

for which Z

(P ) has reducibles. Furthermore σ can be assume to be trans-

verse to the projection p : Z(P ) → P

× I

. Thus Z

(P ) deﬁnes a smooth

compact oriented cobordism betwee n Z

(P ) and Z

(P ). This proves

the invariance of #Z

π,g

(P ) in this case.

This argument extends to the cases b

(Y ) = 0, 1 provided (π

, 0) and

(π

, 1) can be connected by a path which missed the ‘bad’ sets N , W of

Sect. 3.1, Lemmma 14 respectively. However this is not generally true, as

we shall describe below.

4.1. The case b

(Y ) = 0.

The argument in the case b

(Y ) > 1 may fail here due to the presence of

a reducible (unique up to gauge) solution in each Z

σ(t)

(P ). The reducible

stratum of Z

(P ) is an arc which under p projects diﬀeomorphically onto I

The path σ may meet the subset N of Prop. 10 and singularities may occur

in Z

. Choose σ to be the path deﬁned by the family {g(t)} and the family of

perturbations {π(t)} = {(t−1)π

+tπ

}, t ∈ I

. Fix θ a ﬂat connection on L.

Writing π(t) = (α(t), ω(t)), in the notation of Sec. 3.1, the reducible solution

up to equivalence in Z

σ(t)

(P ) is given by θ(t) = θ + d

−1

(ω(t)). Furthermore

the associated family of Dirac operators {D

(t)} determine the cohomology

SEIBERG-WITTEN INVARIANTS ... 195

group H

θ(t)

. According to Prop. 11, by an arbitarily small perturbation π we

can make this a transverse family {D

(t)}. This corresponds to deformation

of σ as σ + π which in turn is induced by perturbations (π

+ π, 0) and

(π

+ π, 0) of the end-points of σ. If π is suﬃciently small then #Z

+π

(P )

coincides with #Z

(P ); thus without loss we may absorb π and simply

assume transversality for π = 0. Then spectral ﬂow for {D(t)} = {D

(t)}

occurs at exactly the values of t where σ(t) meets N .

Proposition 16. Assume {D(t)} is a transverse family and σ is transverse

to the projection p : Z

(P ) → P

×I

away from N . Let σ ∩N = {σ(t

)}

i=1

Then for each t

there is a open neighbourhood N

of θ(t

) such that:

(i) Z

(P )\ ∪ N

is a smooth compact 1-manifold with boundary

(ii) Z

(P ) ∩ N

is diﬀeomorphic to the zeros of the map R × R

→ R,

(t, ξ) 7→ tξ

(iii) Z

∗

σ+π

(P ) ∩ N

= 0 × R

and the orientation of 0 × R

is −ε

∂

∂ξ

where

is the sign of the spectral ﬂow of {D(t)} at t

An immediate consequence of this is the formula

σ(1)

(P ) − #Z

σ(0)

(P ) = −SF{D(t)}

where ‘SF’ on the right denotes the total spectral ﬂow as t varies from 0 to

1. Notice that the left-hand term is actually independent of choice of path,

therefore the spectral-ﬂow term only depends on the end-points of the path.

(In fact, it is possible to verify this directly as well.)

To deﬁne an invariant in this case it is necessary to intro duce a counter-

term which should be a function of ω and g which has the same change

as #Z

π,g

(P ) as we cross from one connected component of P

× I

\N to

another. Such a function can be obtained from the spectral invariants of

[APS].

Proposition 17. Assume b

(Y ) = 0. Let (α, ω, g) be given. Let θ be the

unique (up to gauge) ﬂat connection on L and let a be deﬁned by the condi-

tion d

∗

a = 0, da = ω. Deﬁne

ζ(α, ω, g) =

η (d ∗ − ∗ d|

Ω

even

, g)



dim

ker D

θ+a+α

+ η



θ+a+α



32π

(a + α) ∧ d(a + α)

where η denotes the Atiyah-Patodi-Singer spectral invariant of the associated

operator. Then:

(i) ζ(ω, g) lies in Z

8|H

(Y ;Z)|

; if in addition H

(Y, Z) = 0 it lies in Z

(ii) given the path σ as Prop. 16, we have ζ(σ(1)) − ζ(σ(0)) = SF {D(t)}.

196 YUHAN LIM

Thus we see that the combination

π,g

(P ) + ζ(π, g)

deﬁnes a topological invariant in the case b

(Y ) = 0.

Proof of Proposition 17. Every spin-c structure on Y is obtained by ten-

soring a spin structure on Y with a complex line bundle. By a Theorem

of Milnor every spin Y is the oriented spin boundary of an oriented spin

4-manifold X with b

(X) = 0. Every complex line bundle over Y can be

extended over X; therefore we may assume a spin-c structure P

→ X which

induces the given P → Y . We may also assume X to have a metric which

near the boundary which is a product Y ×[0, ε) of an interval and the metric

on Y and with orientation dy ∧ dt.

We may extend the connection θ ove r L(P

) = det(P

) as the connection

Θ, a as ˆa and α as ˆα over X. These extensions can be taken to be products

over Y × [0, ε). The index theorem of [APS] applied to the Dirac operator

Θ+ˆa+ˆα

over X associated to P

gives:

Index D

Θ+ˆa+ˆα

exp



(Θ + ˆa + ˆα)



−



dim

ker D

θ+a+α

+ η



θ+a+α



Here

(Θ + ˆa + ˆα) =

2π

Θ+ˆa+ˆα

and

A is the

A-polynomial in the Pontrjagin classes. On the other hand

consider the signature operator on X. This has index

sig(X) =

L − η(d ∗ − ∗ d|

Ω

even

, g).

L is the Hirzebruch L-polynomial in the Pontrjagin classes. Since

exp



(B)



= 1 +

(B) +

(B) ∧ c

(B) + . . . ,

A = 1 −

+ . . . ,

L = 1 +

+ . . . ,

the above index formulas give

η(d ∗ − ∗ d|

Ω

even

, g) +



dim

ker D

θ+a

+ η(D

θ+a

)



32π

(a + α) ∧ d(a + α)

SEIBERG-WITTEN INVARIANTS ... 197

(Θ) ∧ c

(Θ) −

sig(X) − Index D

Θ+ˆa+ˆα

If Y is an integral homology s phere then L is trivial and we may choose

its extension over X as the trivial bundle; therefore Θ in this case may be

assumed trivial. Furthermore the intersection form on X is then unimodular

so sig(X) is divisible by 8. Thus ζ is an integer. When Y is not an inte-

gral homology sphere then the term

(Θ) ∧ c

(Θ) depends only on the

topological type of the extension of L over X. It can be identiﬁed with the

(Y ;Z)|

-intersection product of the class [c

(Θ)] ∈ H

(X; Z)/torsion

with itself. Therefore ζ takes values in Z

8|H

(Y ;Z)|

The term η(d∗−∗d|

Ω

even

, g) depends continuously on g whereas according

to [APS]

η(D

θ+a+α

) jumps by the spectral ﬂow. Thus ζ has the correct

behaviour as we cross components of P

× I

\N , as claimed. 

Proof of Proposition 16. The proof of the proposition relies on a detailed

understanding of the Kuranishi local mo del at the singular points [θ(t

)].

Without loss of generality we assume that there is only one value t = t

where {D(t)} is singular. The Seiberg-Witten section which gives Z

(P ) is

of the form ˆs : Ω

(iR) × C(P ) × R → Ω

(iR) ⊕ Γ

(S),

ˆs(η, A, Φ, t) =



∗

−

(Φ, Φ) − ω(t)), D

g(t)

A+α(t)

Φ − ηΦ



where the ‘t’ in the notation denotes a dependence on t. (Note: just as in

Sec. 3 we work with a ﬁxed P → Y with respect to a basepoint metric.)

The linearization of ˆs at η = 0, A = θ(t

), Φ = 0, t = t

is given by

dˆs(δη, δa, δφ, δt) =



∗

(d(δa) + ω

)δt), D(t

)δφ



Let X

θ(t

),0

be the slice of the gauge group action on C(P ) at (0, θ(t

), 0)

(Sec. 2). Then

ker (dˆs) ∩ (Ω

(iR) × X

θ(t

),0

× R) = H

(iR) ⊕ H

θ(t

)

⊕ R

coker (dˆs) = H

θ(t

)

Here R

= span{(d

−1

(ω

)), 1)} in the Ω

(iR) ⊕ R factor. The Kuranishi

obstruction map then takes the form

Ξ : iR × H

θ(t

)

× R

⊃ U → H

θ(t

)

This gives ˆs

−1

(0)/G near (0, θ(t

), 0, t

) as Ξ

−1

(0)/S

. A direct veriﬁcation

shows that Ξ

−1

(0) ⊃ (iR × 0 × R

) ∩ U . The subset (iR × 0 × 0) ∩ U consists

of ‘virtual’ Seiberg-Witten solutions and thus should be ignored to get the

Seiberg-Witten moduli space proper (Lemma 3). The subset (0×0×R

)∩U

are the reducible solutions near (0, θ(t

), 0, t

). Our assumption on σ being

transverse to p away from W means that the closure of the irreducible part

198 YUHAN LIM

of Ξ

−1

(0)/S

, is a compact 1-manifold with boundary except possibly at

(0, 0, 0). Furthermore Ξ

−1

(0) ∩ U ∩ (R × 0 × R

− (0, 0, 0)) = ∅.

By construction the derivative of Ξ at (0, 0, 0) is the zero map. We aim

to compute the second derivative: This will give us the quadratic approxi-

mation to Ξ which will be suﬃcient for our purposes. In the following, we

identify R

with R via t 7→ (d

−1

(ω

)), 1)t.

Claim 18. The second derivative of Ξ at (0, 0, 0) is given by

Ξ(δη, δφ, δt) = cδtδφ − δηδφ

where c is a non-zero real constant and has the same sign as that of the

spectral ﬂow of {D(t)} at t = t

To prove the claim: the obstruction map Ξ is constructed as a c omposition

of the form x 7→ Π ◦ ˆs(x + f(x)) where x ∈ O ⊂ iR × H

θ(t

)

× R

and

f : O → (iR × H

θ(t

)

× R

)

⊥

is given by the implicit function theorem. As

such its derivative at 0 is the zero map. It is then seen that D

Ξ is given

by Π ◦ D

ˆs. This is given by the expression

Ξ(δη, δφ, δt) = Π



)(δtδφ)



− δηδφ.

The map δφ 7→ Π(D

)δφ) deﬁnes a He rmitian transformation on H

θ(t

)

∼

C with respec t to the complex L

-inner product. Thus it is multiplication

by a real constant c = hD

(t)v, vi, v being of unit length. Our assumption

on σ was that at t = t

a single eigenvalue λ(t), |t−t

| < δ, for D(t) changed

from negative to positive or vice-versa. In the ﬁrst case the spectral ﬂow is

+1 and in the latter −1. We have a 1-parameter family of unit eigenvectors

v(t), |t − t

| < δ, such that

D(t)v(t) = λ(t)v(t).

Diﬀerentiating this equation at t = t

and taking the inner product with

v(t

) we obtain using self-adjointness of D(t),

)v(t

), v(t

)i = λ

Thus the sign of the spectral ﬂow is seen to be same as that of λ

). This

proves the Claim.

To continue the proof of Proposition 16: by the Claim, Ξ(η, φ, t) is ap-

proximated up to second order by

(ct − η)φ.

The zeros of the quadratic approximation fall into two branches: 0 × 0 × R

and 0 × H

θ(t

)

× 0. We claim that these two branches gives a complete

picture of the zeros of Ξ near (0, 0, 0). To get a clearer picture, consider

restricting the φ variable to the real span of a ﬁxed non-zero vector in H

Call this map

Ξ. Then Ξ

−1

(0)/S

−1

(0)/ ± 1. Since

Ξ(η, 0, t) = 0 we

may factor out the branch {φ = 0} by setting

Ξ(η, φ, t) = φ.Θ(η, φ, t). Using

SEIBERG-WITTEN INVARIANTS ... 199

Claim 18 the linearization of Θ is seen to be dΘ(δη, δt, δφ) = (cδt − δη). We

have ker dΘ = {δt = δη = 0}. Invoking the Implicit Function Theorem

we see that near (0, 0, 0), Θ

−1

(0) is a smooth arc tangent to {φ = 0} at

(0, 0, 0). This dem onstrates the claimed local structure near the singular

point. For later we note the following: the implicit function theorem gives

a map G : H

θ(t

)

→ R

= {0} × R

⊂ iR × R

such that the closure of the

irreducible part of Ξ

−1

(0) is given by graph(G) = {(0, φ, G(φ)) | φ ∈ H

θ(t

)

What remains is to determine the orientation of the above arc of irre-

ducible solutions. We have an orientation of det Ind{L

η,A,Φ,t

} determined

by that of det Ind{L

η,A,Φ

} according to the map h of (3). (See discussion

following there.) Since we are working at a point where Φ = 0, the orienta-

tion of det Ind{L

η,A,Φ

} is determined by the kernel and cokernel of L

θ(t

),0

;

namely H

(iR)⊕H

θ(t

)

. The long e xact sequence inducing h takes the form

0 → H

(iR) ⊕ H

θ(t

)

→ ker L

θ(t

),0,t

→ R → H

(iR) ⊕ H

θ(t

)

→ coker L

θ(t

),0,t

→ 0.

The map κ in the sequence sends the subspace R

isomorphically onto the

the target space. This isomorphism sends (1, d

−1

(ω

)))r 7→ r, r ∈ R.

Choose an orientation of coker L

θ(t

),0

= H

(iR) ⊕ H

θ(t

)

; then s ince β in

the sequence is an isomorphism, our orientation convention dictates that

coker L

θ(t

),0,t

has the orientation induced by β, and the orientation on

ker L

θ(t

),0,t

is the product orientation ker L

θ(t

),0

⊕R

, where R

is oriented

via κ.

In order to determined orientations in the local Kuranishi picture cor-

rectly we shall need to combine the obstruction map Ξ with a local slice

condition coming from the S

-action, which is the inverse of complex mul-

tiplication on the H

θ(t

)

factor. The set graph(G) ⊂ Ξ

−1

(0) represents

the closure of the S

-orbits of the irreducible solutions near (0, 0, 0). Let

0 6= v ∈ H

θ(t

)

. Then the linearization of the S

-action at (0, v, G(v)) is

a map iR → iR × H

θ(t

)

× R

, δγ 7→ (0, −(δγ)v, 0). The adjoint of this

map sends (δη, δφ, δt) 7→ −ihiv, δφi. Therefore a further local description

for Ξ

−1

(0)/S

near (0, v, G(v)) is the zeros of the map

χ : iR × H

θ(t

)

× R

→ iR × H

θ(t

)

, χ(η, φ, t) = (−ihiv, φi, Ξ(η, φ, t)).

In what follows, we may for simplicity assume G = 0; the result for general

G is obtained by working suﬃciently close to (0, 0, 0) where graph(G) is

approximated to arbitarily high order by 0 × H

θ(t

)

× 0. With this assumed,

the irreducible zeros of χ is the set of positive multiples of (0, v, 0). The

normal bundle to 0×H

θ(t

)

×0 at (0, 0, 0) is iR×0 ×R

. This is mapped via

dΘ isomorphically onto H

θ(t

)

. If we pull-back the complex orientation by

dΘ, then the induced orientation on iR×0×R

is cδη∧δt. By continuity this

is carried to the point v as the same orientation. The last remaining direction

200 YUHAN LIM

to the normal bundle of χ

−1

(0) at v is given by (0, iv, 0). This is mapped by

dχ to (−ihiv, ivi, 0) = (−i, 0). Let us take the product orientation on iR ×

θ(t

)

for the range space of χ, in this order (the ﬁnal answer is independent

of this choice); then our orientation convention dictates that the domain

space of χ is oriented in the order iR × H

θ(t

)

× R

. Then the pull-back

of the orientation on iR × H

θ(t

)

to the normal bundle of χ

−1

(0) at v is

−δθ ∧ cδη ∧ δt, where δθ is the (0, iv, 0) direction. Let δr be the direction

given by v, and εδr the induced orientation of χ

−1

(0) near v. Then we

require that the orientation on χ

−1

(0) followed by the orientation in the

normal direction equals the orientation on iR × H

θ(t

)

× R

, that is

εδr ∧ (−δθ ∧ cδη ∧ δt) = δη ∧ δr ∧ δθ ∧ δt.

This shows the induced orientation on χ

−1

(0) as −cδr, as claimed.

4.2. The case b

(Y ) = 1.

This case is similiar but with some slight diﬀerences to b

(Y ) = 0. Any

path σ which connects (π

, g

) to (π

, g

) in P

× I

may cross the codi-

mension 1 subset W. At the points where σ meets W the corresponding

σ(t)

(P ) will admit an S

’s worth of reducibles, otherwise Z

σ(t)

(P ) contains

no reducibles. As following our notation conventions, σ(t) in components is

(π(t), g(t)) or (α(t), ω(t), g(t)).

We may by general transversality arguments assume that σ meets W

transversely and orthogonally and transverse to the projection p : Z

∗

(P ) →

×I

away from W. Let {t

} b e the ﬁnite set of values for which σ(t

) ∈ W.

To simplify matters even more, since W the preimage of a set in P

we may

assume near W that σ lies in the subset P

× {t

}. Hence for values near t

the metric represented by σ is unchanging.

We can always ﬁnd connections A

such that F

= ω(t

). Using the

value α(t

), we can as in Sec. 3.1 form the 1-parameter family of operators

(s)}. By Prop. 11 we can make this family transverse by an arbitarily

small perturbation π

. This perturbation can be achieved by a perturbation

of σ, supported for values of t near t

, and maintaining the original properties

of σ. Thus we can assume {D

(s)} is a transverse family and we drop the

‘0’ subscript notation. Finally let s

i,j

be the values of s for which {D

(s)}

has spectral ﬂow. Denote by A

i,j

the connection A

+ s

i,j

A technical issue which will be signiﬁcant is the orientation of the family

(s)}. Looking back at the deﬁnition in Sec. 3.1 we see that this involved

a certain choice of a non-zero element a

in H

(iR). We shall make a

speciﬁc choice for each i. The assumption that σ meets W orthogonally

means in particular that the derivative ω

) is L

-orthogonal to the exacts.

Thus d

∗

(ω

)) = 0 so ∗ω

) is closed. For a chosen i we now make the

convention that the a

should be a positive multiple of [− ∗ ω

)].

SEIBERG-WITTEN INVARIANTS ... 201

Proposition 19. For each (i, j) there is a open neighbourhood N

i,j

of [A

i,j

]

such that (i) Z

(P )\ ∪ N

i,j

is a smooth compact 1-manifold with boundary

(ii) N

i,j

is diﬀeomorphic to the zeros of the map R × R

→ R, (s, ξ) 7→ sξ

with Z

∗

(P ) ∩ N

i,j

= 0 × R

(iii) the orientation of 0 × R

is ε

i,j

∂

∂ξ

where

i,j

is the sign of the spectral ﬂow of {D

(s)} at s

i,j

, as s varies from 0 to 1.

Proof. This largely proceeds in the manner of the cas e b

(Y ) = 0. We

continue to use notation introduced there. Again, without loss, we may

assume σ meets W exactly once, at t = t

. We consider again the map ˆs of

the case b

(Y ) = 0. Computed at η = 0, A = A

1,j

, Φ = 0, this time we ﬁnd

ker (dˆs) ∩ (Ω

(iR) × X

1,j

× R) = iR ⊕ H

(iR) ⊕ H

1,j

coker (dˆs) = H

1,j

We have the obstruction map Ξ : iR × H

(iR) × H

1,j

→ H

1,j

whose

second derivative at (0, 0, 0) is

Ξ(δη, δa, δφ) =

Π(δa · δφ) − δηδφ.

Then if we let δa = − ∗ ω

)δt, the term

Π(δa · δφ) =

δtΠ(− ∗ ω

) · δφ) = cδtδφ

where c is a non-zero real constant with the same sign of the spectral ﬂow

(s)} at s

1,j

. As in the b

= 0 case, the irreducible zeros of Ξ are mo delled

by the subset 0 × 0 × H

1,j

and the reducible zeros by 0 × H

(iR) × 0.

Let us now deal with the orientations in this case. Looking at the long

exact sequence inducing h of (3) we see

0 → H

(iR) ⊕ H

1,j

→ ker L

1,j

,0,t

→ R

→

(iR) ⊕ H

1,j

→ coker L

1,j

,0,t

→ 0.

We note that κ(t) = −ω

)t maps isomorphically onto the H

(iR) factor,

and coker L

1,j

,0,t

is H

(iR) ⊕ H

(iR). β is the obvious projection. Let us

assume the canonical orientations on H

(iR), H

1,j

, and the orientation on

(iR) induced by κ, which is given by −∗ω

). Finally choose the product

orientation (in the order indicated) on H

(iR) ⊕ H

1,j

. Then

ker L

1,j

,0,t

is identically oriented and coker L

1,j

,0,t

is oriented according

to the order H

(iR) ⊕ H

1,j

Let v ∈ H

1,j

. Then combining the slice condition with Ξ gives the moduli

space near (0, 0, v) (as before we may assume G = 0) as the zeros of the map

χ : iR × H

(iR) × H

1,j

→ iR × H

1,j

χ(η, a, φ) = (−ihiv, φi, Ξ(η, a, φ)).

202 YUHAN LIM

As before the zeros of χ are the positive multiples of (0, 0, v). The pull-back

of the orientation on the target space onto the normal bundle of χ

−1

(0) is

given by −δθ ∧cδη ∧δa where δθ is the angular coordinate on H

1,j

. Letting

r be the direction determined by v and εδr the induced orientation, then we

require

εδr ∧ (−cδθ ∧ δη ∧ δa) = δη ∧ δa ∧ δr ∧ δθ

which gives the induced orientation on χ

−1

(0) as −cδr. 

As mentioned b efore, Z

σ(t)

(P ) admits reducible solutions exactly σ(t) ∈

W. This corresponds to when [

2π

ω(t)] coincides with the c lass c

(L)

. Let

U denote a connected component of H

(Y ; R)−{c

(L)

}. Then if our path

σ has the property that [

2π

ω(t)] ∈ U for all t, then #Z

σ(0)

(P ) = #Z

σ(0)

Therefore #Z

α,ω,g

(P ) is an integer-valued function depending only on the

choice of U. Denote this function as τ(U).

We think of {c

(L)

} as a ‘wall’ in H

(Y ; R). Then as we cross this wall

τ changes. This change can be determined from the previous proposition to

give a ‘wall-crossing’ formula.

Corollary 20. Let a ∈ H

(Y ; Z)/torsion be an indivisible class and let

(L)

= 2na. Let U

be the component of H

(Y ; R)−{c

(L)

} containing

(2n ± 1/2)a. Then

τ(U

) − τ(U

−

) = n.

Proof. Take (π

, g

) and (π

, g

) which deﬁne the values τ(U

) and τ (U

−

)

respectively. Choose our connecting path σ with properties as used for as

for Prop. 19. Without loss, We may suppose that σ cros ses W exactly once,

say at t = t

. We now follow the notation and ideas in the proof of Prop. 19.

According to Prop. 19 We need then to compute the total spectral ﬂow of

the family {D

(s)} as s varies from 0 to 1. The orientation of this family

is determined by − ∗ ω

). We shall choose a to be consistent with this

orientation, but the statement of the corollary is actually independent of

this choice. Take a positive multiple ω of ω

(t) such that with [

2π

ω] = 2a.

Thus A

− ∗ωs, 0 ≤ s < 1 parameterizes all the reducibles in Z

σ(t

)

(P ).

We may deform the family {D

(s)} preserving self-adjointness to the family

g(t

)

−∗ωs

}, s ∈ [0, 1]. Thus it suﬃces to compute the spectral ﬂow for this

family. Notice that there is a gauge transformation g such that g(A

) = A

−

∗ω, or equivalently g

−1

dg = −∗ω. A theorem of [APS] says that the spectral

ﬂow of the Dirac operators {D

g(t

)

−∗ωs

} is equivalent to computing the index

of a Dirac operator D

(4)

on Y ×S

with a spin-c structure obtained by taking

the product P ×[0, 1] over Y ×[0, 1] and identifying via g : P ×{1} → P ×{0}.

A is a connection which is in temporal gauge and coincides with A − ∗ωs on

L × {s}. (Remark: We follow the orientation c onventions of [APS] closely.

SEIBERG-WITTEN INVARIANTS ... 203

In particular Y × S

has the product orientation dy ∧ ds where dy is the

orientation form on Y and s the real coordinate on S

thinking of it as

R/Z.) Denote the resulting dete rminant on Y × S

by L

. The index of

(4)

is given by

), [Y × S

]i +

sig(Y × S

To compute the ﬁrst term, we notice F

= d(A

−∗ωs) = F

−∗ωds. Then

), [Y × S

]i =

Y ×[0,1]

2π

∧

2π

= −

4π

Y ×[0,1]

− ∗ωds) ∧ (F

− ∗ωds)

= −

4π

(−2)

Y ×[0,1]

∧ ∗ωds

= −

4π

(−2)

Y ×[0,1]

nω ∧ ∗ωds

= −

4π

(−2n)

ω ∧ ∗ω

= −

4π

(−2n)4πia · PD(4πia)

= 8n.

Here ‘PD’ denotes Poincare Duality. Since sig(Y × S

) = 0, the index of

(4)

is n and the corollary follows. 

References

[APS] M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian

geometry I, II, III, Math. Proc. Camb. Phil. Soc., 77 (1975), 43-69; 78 (1975),

405-32; 79 (1976), 71-99.

[B] J-P. Bourgiugnon, Spinors, Dirac operators, and changes of metric, in Diﬀerential

Geometry: Geometry in mathematical physics and related topics (Los Angeles CA,

1990), 41-44, Proc. Sympos. Pure Math., 54, part 2, Amer. Math. Soc., Providence,

RI, 1993.

[BG] J-P. Bourguignon and P. Gauduchon, Spineurs, operateurs de Dirac et variations

de metriques, Comm. Math. Phys., 144(3) (1992), 581-599.

[Ch] W. Chen, Casson invariant and Seiberg-Witten gauge theory, Turkish J. Math., 21

(1997), 61-81.

[D] S.K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bull.

A.M.S., 33(1) (1996), 45-70.

[Fy] K.A. Froyshov, The Seiberg-Witten equations and 4-manifolds with boundary,

Math. Res. Lett., 3 (1996), 373-390.

204 YUHAN LIM

[H] O. Hijazi, A conformal lower bound for the smallest eigenvector of the Dirac oper-

ator and Killing spinors, Comm. Math. Phys., 104(1) (1986), 151-162.

[KM] P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective

plane, Math. Res. Let., 1(6) (1994), 797-808.

[LM] H.B. Lawson and M-L. Michelsohn, Spin Geometry, Princeton Univ. Press, 1989.

[L] Y. Lim, The Equivalence of Seiberg-Witten and Casson Invariants for Homology

3-spheres, Math. Res. Lett., 6(6) (1999), 631-643.

[Mar] M. Marcolli, Seiberg-Witten-Floer theory and Heegaard splittings, Int. J. Math.,

7(5) (1996), 671-696.

[MT] G. Meng and C.H. Taubes, SW = Milnor Torsion, Math. Res. Lett., 3(5) (1996),

661-674.

[M] J. Morgan, The Seiberg-Witten equations and applications to the topology of smooth

four-manifolds, Princeton Math. Notes, 44, Princeton Univ. Press, 1996.

[N] L. Nicolaescu, Lattice points, Dedkind-Rademacher sums and a conjecture of

Kronheimer-Mrowka, preprint.

[T] C.H. Taubes. Casson’s invariant and gauge theory, J. Diﬀ. Geom., 31 (1990), 547-

599.

Received February 1, 1998 and revised May 14, 1999.

University of Ca lifornia

Santa Barbara, CA 93106

E-mail address: ylim@gauss.math.ucsb.edu

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

VIRTUAL HOMOLOGY OF SURGERED TORUS BUNDLES

Joseph D. Masters

Let M be a once-punctured torus bundle over S

with

monodromy h. We show that, under certain hypotheses on

h, “most” Dehn-ﬁllings of M (in some cases all but ﬁnitely

many) are virtually Z-representable. We apply our results to

show that even surgeries on the ﬁgure eight knot are virtually

Z-representable.

1. Introduction.

Embedded incompressible surfaces are fundamental in the study of 3-mani-

folds. Accordingly, the following conjecture of Waldhausen and Thurston

has attracted much attention:

Conjecture 1.1. Let M be a closed, irreducible 3-manifold with inﬁnite π

Then M has a ﬁnite cover which is Haken.

The focus of this paper is the following, stronger, conjecture:

Conjecture 1.2. Let M be as above. Then M has a ﬁnite cover

M with

(

M, Z) inﬁnite.

If M is a compact 3-manifold, we say that M is Z-representable if

(M, Z) is inﬁnite. If M satisﬁes the conclusion of Conjecture 1.2, we

say that M is virtually Z-representable.

We shall give what appear to be the ﬁrst examples of 3-manifolds with

torus boundary for which all but ﬁnitely many ﬁllings are virtually Z-

representable, but not Z-representable (in fact non-Haken). Boye r and

Zhang have independently given examples of knot complements for which

all but ﬁnitely many ﬁllings are virtually Haken, but non-Haken [BZ].

Before we can state our results, we must establish some notation. Let F

be a once-punctured torus with π

(F ) = h[x], [y]i, and basepoint x

∈ ∂F

(see Fig. 1).

Any orientation-preserving homeomorphism h : F → F is isotopic to one

of the form h = D

· · · D

. Here D

and D

are Dehn twists along

simple closed curves homologous to x and y, respectively. The twists D

205

206 JOSEPH D. MASTERS

Figure 1. Notation for the once-punctured torus bundle M.

and D

induce the following actions on π

(F ):

(x) = x

(y) = yx

(x) = yx

(y) = y.

We may assume h ﬁxes ∂F . Let M

= (F × I)/h be the once-punctured

torus bundle with monodromy h. We specify a framing for H

(∂M

, Z) by

setting the longitude β = ∂F oriented counter-clockwise, and the meridian

α = (x

×I)/h, where x

is some point on ∂F , and α is oriented as in Fig. 1.

Then, for coprime integers (µ, λ), M

(µ, λ) denotes the manifold obtained

by gluing a solid torus to M

in s uch a way that the curve α

becomes

homotopically trivial.

We shall prove:

Theorem 1.3. Let M

be a once-punctured torus bundle over S

, with

monodromy h = D

· · · D

, and let n = g.c.d{s

, . . . , s

}, R =

+ · · · + r

(i) If n is divisible by some m such that m ≥ 6 and m is even or m = 7,

and if |λ| > 1, then all but ﬁnitely many Dehn-ﬁllings M

(µ, λ) are

virtually Z-representable.

(ii) If n is divisible by some m such that m ≥ 5, m is odd, and m 6= 7,

and if 1/|Rµ − λ| + 1/|Rµ − 2λ| + 1/|λ| < 1, then M

(µ, λ) is virtually

Z-representable.

(iii) If n is divisible by 4, and if 2/|Rµ − 2λ| + 1/|λ| < 1, then M

(µ, λ) is

virtually Z-representable.

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 207

Remarks. 1. Analogous results hold if we replace n by gcd{r

, . . . , r

} and

R by s

+ · · · + s

2. It was shown in [B1] that if m ≥ 2, n ≥ 2 and mn ≥ 8 but mn 6= 9,

then all non-integral surgeries are virtually Z-representable. In [B2] it was

shown that if 4|n, then for each µ, M

(µ, λ) is virtually Z-representable for

all but ﬁnitely many λ coprime to µ.

3. From [CJR] and [FH], all but ﬁnitely many surgeries on a once-

punctured torus bundle over S

yield non-Haken manifolds.

Theorem 1.3 may b e used to show that, for certain choices of f , all but

ﬁnitely many surgeries on M

are virtually Z-representable. For example:

Theorem 1.4. Let f = (D

)

. Then every surgery on M

is virtually

Z-representable.

The proof of Theorem 1.4 appears in Section 3.

In order to state the next theorem, we require some notation. Let −1 =

−1

)

, the central involution on the punctured torus. If h is a home-

omorphism of the punctured torus, −h stands for (−1)h.

Theorem 1.5. Let N = M

−D

(also known as “the ﬁgure eight knot’s

sister”). Then if 1/|µ − λ| + 1/|µ − 2λ| + 1/|λ| < 1, N(µ, λ) is virtually

Z-representable.

Theorem 1.6. Let K denote the ﬁgure-eight knot and let M denote S

−K.

Then, with respect to the canonical framing of knots in S

, any surgery of

the form M(2µ, λ) is virtually Z-representable.

Other results on virtually Z-representable ﬁgure-eight knot surgeries may

be found in [M], [KL], [H], [N] and [B3]. In particular, it was shown in [KL]

and [B3] that surgeries of the form M (4µ, λ) are virtually Z-repres entable.

It was also shown in [B3] that surgeries of the form M (2µ, λ) are virtually

Z-representable if λ = ±7µ (mod 15). Finally, it was shown in [Bart] that

every non-trivial surgery of M contains an immersed incompressible surface.

Our techniques are extensions of Baker’s. The main new ingredient is the

use of group theory to encode the combinatorics of cutting and pasting.

I would like to thank Professor Alan Reid for his help and patience.

2. Construction of covers.

We begin by recalling Baker’s construction of covering spaces of M

(µ, λ)

(see [B1], [B2]). Let n be as in the statement of Theorem 1.3, and let

be the kn-fold cover of F associated to the kernel of the map φ : π

(F ) →

× Z

, with φ([x]) = (1, 0) and φ([y]) = (0, 1) (see Fig. 2).

Now create a new cover,

F , of F by making vertical cuts in each row of

F , and gluing the left side of each cut to the right side of another cut in the

208 JOSEPH D. MASTERS

Figure 2. The cover

F of F .

same row. An example is pictured in Figure 3, where the numbers in each

row indicate how the edges are glued.

If h lifts to a map

h :

F →

F , then the mapping cylinder

F /

h is

a cover of M

. Furthermore, if the loop α

lifts to loops in

, then the

cover extends to a cover

(µ, λ) of M

(µ, λ).

If the cover

exists, then we m ay compute its ﬁrst Betti number with

the formula b

(

) = rank (fix(

∗

)), where

∗

is the map on H

(

M, Z)

induced by

h, and f ix(

∗

) is the subgroup of H

(

M, Z) ﬁxed by

∗

(see [H]

for a proof). We shall use this formula to prove that, in some cases, b

(

is greate r than the number of boundary components of

M, which ensures

that b

(

M(µ, λ)) > 0.

We now introduce some notation to describe the cuts of

F (see Fig. 3).

is naturally divided into rows, which we label 1, . . . , n. The cuts divide each

row into pieces, each of which is a square minus two half-disks; we number

them 1, . . . , k. If we slide a point in the top half of the i

row through the

cut to its right, we induce a permutation on {1, . . . , k}, which we denote

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 209

Piece 3, Row 1

1 3 4

5 6

1 2 3 4

5 6

6 2

1 5

2 3

4 5 6

3 6 2 1

4 5

2 6 3 2

4 1

5 4

σ = (15)(2463)

σ = (146235)

σ =

Row 1

Row 2

Row 3

Figure 3. The permutations encode the combinatorics of

the gluing.

. Thus the cuts on

F may be encoded by elements σ

, . . . , σ

∈ S

, the

permutation group on k letters .

Next, we ﬁnd algebraic conditions on the σ

’s which will guarantee that

the cover of F extends to a cover of M(µ, λ). We ﬁrst must pick k, n, and

{σ

, . . . , σ

} so that h lifts to

F .

Lemma 2.1. If

I. [σ

, σ

· · · σ

i−1

] = 1 for all i and

II. σ

· · · σ

= 1

then h lifts to

F .

Proof. Note that D

lifts to Dehn twists on

F . Therefore, we need only

ensure that D

lifts. We shall attempt to lift D

to a sequence of “fractional

Dehn twists” along the rows of

F . Let ˜x

denote the disjoint union of the

lifts of x to the i

row of

F . We ﬁrst attempt to lift D

to row 1, twisting

1/k

of the way along ˜x

. Considering the action on the bottom half of row

1, we ﬁnd that the cuts are now matched up according to the permutation

−1

. Thus, for D

to lift to row 1 we assume σ

and σ

commute.

We now twist along ˜x

. The top halves of the squares in row 2 are moved

according to the permutation σ

, and the lift will extend to all of row 2 if

and only if σ

commutes with σ

. We continue in this manner, obtaining

the conditions in I. After we twist through ˜x

, we need to be back where we

started in row 1, so we require the additional condition σ

· · · σ

= 1. 

Note that the loop α

lifts homeomorphicly to loops in

= id, and

that the loop β

lifts to loops in

if (σ

i+1

−1

)

= id for all i = 1, . . . , n.

Then, by considering the action of

h on

, the following condition for a

loop in ∂M

to lift to

is easily veriﬁed:

210 JOSEPH D. MASTERS

Lemma 2.2. The loop α

⊂ ∂M

lifts homeomorphicly to loops in

if and only if

III. (σ

· · · σ

)

Rµ

(σ

i+1

−1

)

= 1, for i = 1, . . . , n.

Therefore we may construct covers of M

(µ, λ) simply by ﬁnding permu-

tations satisfying conditions I-III.

Proof of Theorem 1.3.

Case 1. m = 4.

Construction of the cover of M

(µ, λ).

To construct a cover of M

(µ, λ), we must ﬁrst construct a cover of F .

It was shown in the discussion prior to Lemma 2.1 that there is a unique

such cove r associated to any four permutations σ

, σ

and σ

in any

permutation group S

To ensure that the cover of F extends to a cover of M

, we shall set

= σ

−1

and σ

= σ

−1

(see Fig. 4a). Then conditions I and II of Lemma

2.1 are satisﬁed automatically, so that any choice of σ

and σ

will determine

a cover of M

To ensure that the cover extends to M

(µ, λ), we must arrange for the

surgery curve α

to lift to

. By Lemma 2.2, α

will lift provided

that σ

, . . . , σ

satisfy condition III, which reduces to:

Rµ−2λ

= 1(1)

(σ

)

= 1(2)

Rµ−2λ

= 1(3)

(σ

)

= 1.(4)

Any pair of permutations σ

and σ

satisfying Equations (1)-(4) deter-

mines a unique cover of M

(µ, λ). We now turn our attention to the con-

struction of such permutations.

Consider the abstract group G generated by the symbols ¯σ

and ¯σ

, satis-

fying relations (1)-(4). G is a (|Rµ − 2λ|, |Rµ − 2λ|, |λ|)-triangle group. It is

well-known that if 1/|Rµ−2λ|+1/|Rµ−2λ|+1/|λ| < 1, then G is residually

ﬁnite, and hence surjects a ﬁnite group H such that the images of ¯σ

, ¯σ

and ¯σ

¯σ

have order |Rµ − 2λ|. By taking the permutation representation

of H, we then obtain permutations σ

and σ

satisfying conditions (1)-(4).

Note that the permutations act on |H| letters, so

M is a 4|H|-fold cover of

Associated with the permutations σ

and σ

we have covers

and

F of

and F , and a cover

(µ, λ) of M

(µ, λ);

Claim. b

(

(µ, λ)) > 0.

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 211

−1

Figure 4

a. The cover when n = 4. b. The cover when n = 5.

Proof of claim. It suﬃces to show that

∗

has a non-peripheral class [δ] ∈

(

F ) with

∗

([δ]) = [δ]. To construct this element, we shall ﬁrst ﬁnd a

non-peripheral class [δ

] in row 2, as follows.

−1

Figure 5. The surface

(with |H| = 4).

Consider the sub-surface

obtained by deleting rows 1, 3 and 4 from

(see Fig. 5). The punctures of

are in 1-1 correspondence with the cycles

212 JOSEPH D. MASTERS

of σ

, σ

and σ

. Any permutation τ coming from the permutation repre-

sentation of H decomposes as a product of |H|/order(τ) disjoint order(τ )-

cycles. Therefore

has |H|(1/order(σ

) + 1/order(σ

)) <

|H| punctures. Since χ(

) = −|H|, we deduce that

contains a non-

peripheral class [δ

]. The class δ

also represents a non-peripheral class in

F , since it has non-zero intersection number with a class of

F in row 2.

We may ﬁnd a corresponding non-peripheral loop δ

in row 4, such that

I([δ

+ δ

], [˜y

]) = 0 for all i (see Fig. 6 for the notation and the idea of

the proof). Let [δ] = [δ

+ δ

]. Then, since [δ] has non-zero intersection

number with classes in row 2 and row 4, it is a non-peripheral class. We

have I[δ, ˜y

] = 0 for all i (where I(., .) denotes oriented intersection number);

therefore [δ] is ﬁxed by

∗

, and since

ﬁxes rows 2 and 4, it is ﬁxed by

∗

. Therefore it is ﬁxed by

∗

, concluding the proof of the claim, and of

Case 1.

Case 2. m ≥ 5 and m is odd.

Case 2a. m = 5.

The construction proceeds analogously to the case m = 4. We require per-

mutations σ

, . . . , σ

satisfying conditions I-III. Again, to simplify matters,

we shall impose some extra conditions: σ

= id, σ

= σ

−1

, and σ

= σ

−1

(see Fig. 4). Then conditions I-III reduce to:

Rµ−λ

= 1

(σ

)

= (σ

)

= 1

Rµ−2λ

= 1.

Again, these relations determine a triangle group, which, under the hypothe-

ses on µ and λ, is hyperbolic. The rest of the proof is identical to Case 1,

except that now the ﬁxed class is in rows 3 and 5.

Case 2b. m ≥ 9 and m is odd.

Consider the cover obtained by setting σ

= σ

−1

, σ

= id, σ

= σ

−1

= σ

, σ

= σ

−1

, and for i = 4, . . . , k, σ

2i+1

= σ

−1

(see Fig. 7a). 

The corresponding relations are:

Rµ−2λ

= 1(5)

Rµ−λ

= 1(6)

(σ

)

= (σ

)

= 1(7)

(σ

)

= 1(8)

Rµ−2λ

= 1 for i = 4, . . . , k(9)

(σ

2i+2

)

= 1 for i = 4, . . . , k − 1(10)

(σ

)

= 1.(11)

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 213

−1

row 4

row 2

row 4

row 2

−1

Figure 6. How to ﬁnd cancelling loops in rows 2 and 4.

These relations again determine a Coxeter group. It is well-known (see

[V]) that any such group surjects a ﬁnite group “without collapsing”– i.e.,

such that the orders of the images of the σ

’s and σ

’s are as given in

(5)-(11). Then, arguing as in Case 1, we may ﬁnd a non-peripheral ﬁxed

class in rows 2 and 5.

214 JOSEPH D. MASTERS

−1

2k-1

a b

Figure 7. a. The cover for n = 2k + 1 ≥ 9. b. The cover

for n = 2k ≥ 8.

Case 3. n = 6

Case 3a. 2/|Rµ − λ| + 1/|λ| < 1.

Again, we need permutations σ

, . . . , σ

satisfying I-III. In this case we

impose the additional conditions σ

= id, σ

= σ

−1

, σ

= id, and σ

= σ

−1

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 215

Then conditions I-III reduce to:

Rµ−λ

= 1

(σ

)

= (σ

)

= 1

Rµ−λ

= 1.

These relations dete rmine a triangle group, and we ﬁnd a ﬁxed class in

rows 3 and 6.

Case 3b. |λ| > 2 and |Rµ−3λ| ≥ |λ|, or λ is even (non-zero) and |Rµ−3λ| ≥

When n = 3, conditions I-I II may be abelianized to obtain a cyclic

group of order |Rµ − 3λ|. Speciﬁcally, they are satisﬁed by setting σ

(1, 2, . . . , Rµ − 3λ), σ

= σ

−2

, and σ

= σ

. For n = 6, we may “double”

this cover: That is take σ

, σ

as above, and then set σ

= σ

, σ

= σ

and σ

= σ

. Then we modify the corresponding cover

M(µ, λ) of M(µ, λ)

by making horizontal cuts in adjacent squares of row 3 and gluing the ﬂaps

back together as indicated by Fig. 8. If λ is even, we make two non-adjacent

cuts and glue the top of one to the bottom of the other. If λ is odd, we

make (|λ|−1)/2 pairs of adjacent cuts and glue the top of the one cut to the

bottom of the other cut in its pair. Now make the same cuts in row 6, with

the same identiﬁcations. Since rows 3 and 6 are ﬁxed by

, D

still lifts

to the modiﬁed

(µ, λ), and since the ˜y’s still project 6 to 1 onto y, D

lifts also; so h lifts. Also, one may check that α

still lifts, so

(µ, λ)

remains a cover of M

(µ, λ).

To see that b

(

(µ, λ)) > 0, note that

ﬁxes rows 3 and 6, so it is

enough to ﬁnd a non-peripheral loop in row 3 and add it to the corresponding

loop in row 6 with opposite orientation. As in Case 1, the existence of such

a non-peripheral loop follows from an Euler characteristic argument (or see

Fig. 8).

Note that Case 3a or 3b applies to all but ﬁnitely many (µ, λ) with |λ| > 1.

Case 4. n = 2k ≥ 8.

Case 4a. 2/|Rµ − 2λ| + 1/|λ| < 1. Set σ

= σ

−1

, σ

= σ

−1

, σ

= σ

−1

, and σ

= σ

−1

2i−1

for i = 4, . . . , k (see Fig. 7b). Then, as in Case

2, these relations determine a Coxeter group. We may ﬁnd a non-peripheral

ﬁxed class in rows 2 and 4.

Case 4b. |Rµ − λ| ≤ 2 We cannot guarantee, in this case , that there will

always be a cover with b

> 0, but we shall show that there are at most

ﬁnitely many exceptions.

We argue as in Case 3b. Take permutations σ

, . . . , σ

, and consider the

relations obtained by abelianizing conditions I-III. We claim that they can

be satisﬁed by setting σ

= (1, 2, 3, . . . , N), for some N, and setting each

216 JOSEPH D. MASTERS

1 2 3 4

Figure 8. a. The cover and ﬁxed class for n = 6, Rµ − 3λ =

4, λ = 3. b. The cover and ﬁxed class for n = 6, Rµ − 3λ =

4, λ = 2.

to an appropriate power of σ

. We have already seen that this may be

done when k = 3.

The σ

’s must s atisfy the following c onditions:

Rµ−λ

= 1(12)

Rµ

Rµ−λ

= 1(13)

Rµ

· · · σ

Rµ

k−2

(14)

Rµ−λ

k−1

= 1(15)

Rµ+λ

Rµ

· · · σ

Rµ

k−1

(16)

Rµ−λ

= 1(17)

· · · σ

= 1.(18)

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 217

We shall assume that this system has a cyclic solution, so we may sub-

stitute σ

= σ

. Then, Equations (12)-(18) are e quivalent to the following

conditions on the exponents (all of the following equations in this case are

taken mod N):

Rµ − λ + λe

= 0(19)

Rµ + (Rµ − λ)e

+ λe

= 0(20)

Rµ + Rµe

+ · · · + Rµe

k−2

+ (Rµ − λ)e

k−1

+ λe

= 0(21)

Rµ + λ + Rµe

+ · · · + Rµe

k−1

+ (Rµ − λ)e

= 0(22)

1 + e

+ · · · + e

= 0(23)

(22) and (23) imply that λ = λe

. Let us set e

= 1, eliminating Equation

(22). Then, using (23), we may pair oﬀ (19) and (21) to deduce that λe

λe

k−1

, and we set e

= e

k−1

to eliminate (21). Similarly, we set e

= e

k−2

and so on. If k is even, we are left with equations:

Rµ − λ + λe

= 0(24)

Rµ + (Rµ − λ)e

+ λe

= 0(25)

Rµ + Rµe

+ · · · + (Rµ − λ)e

k/2−1

+ λe

k/2

= 0(26)

Rµ + Rµe

+ · · · + (Rµ − λ)e

k/2

+ λe

k/2

= 0(27)

2 + 2e

+ · · · + 2e

k/2

= 0.(28)

If we replace (28) by

1 + e

+ · · · + e

k/2

= 0(29)

then we may eliminate (27). Then solve for λe

, λ

, . . . , λ

k/2−1

k/2

. By

(29), we have:

k/2−1

+ λ

k/2−2

(λe

) + λ

k/2−3

(λ

) + · · · + λ

k/2−1

k/2

= 0.(30)

Substituting our solutions for λe

, λ

and so on, we get the equation N = 0

for some integer N; the system has a solution in Z/NZ.

If k is odd, then our reduced system looks like:

Rµ − λ + λe

= 0(31)

Rµ + (Rµ − λ)e

+ λe

= 0(32)

218 JOSEPH D. MASTERS

Rµ + Rµe

+ · · · + (Rµ − λ)e

(k−1)/2

+ λe

(k+1)/2

= 0(33)

Rµ + Rµe

+ · · · + (Rµ − λ)e

(k+1)/2

+ λe

(k−1)/2

= 0(34)

2 + 2e

+ · · · + 2e

(k−1)/2

+ e

(k+1)/2

= 0.(35)

Adding (33) and (34) gives a multiple of (35), so we may eliminate (34).

Then we s olve for λe

, λ

, . . . , λ

(k−1)/2

(k+1)/2

. By (35), we have:

2λ

(k−1)/2

+ 2λ

(k−3)/2

(λe

) + 2λ

k−5/2

(λ

) + · · ·

+2λ(λ

(k−3)/2

(k−1)/2

) + λ

(k−1)/2

(k+1)/2

= 0.

And again we get a solution in Z/NZ for some N.

Then, as in Case 3b, M(µ, λ) will have a cover with b

> 0, provided that

|N| ≥ |λ| and |λ| > 2. Solving for N, if k is even, gives:

N = λ

k/2−1

+ λ

k/2−2

(λ − Rµ) + λ

k/2−3

[(λ − Rµ)

− Rµλ](36)

+λ

k/2−4

[(λ − Rµ)((λ − Rµ)

− Rµλ) − Rµλ(λ − Rµ) − Rµλ

] + · · ·

and if k is odd:

N = 2λ

(k−1)/2

+ 2λ

(k−3)/2

(λ − Rµ) + 2λ

(k−5)/2

[(λ − Rµ)

− Rµλ](37)

+2λ

(k−7)/2

[(λ − Rµ)((λ − Rµ)

− Rµλ) − Rµλ(λ − Rµ) − Rµλ

]

+ · · · + 1[..].

We are supposing that |Rµ − λ| ≤ 2. So for large µ or λ, Rµ/λ → 1, and

for k even,

N = o[λ

k/2−1

+ λ

k/2−3

(−λ

) + λ

k/2−4

(−λ

) + · · · ]

= o[λ

k/2−1

− λ

k/2−1

− λ

k/2−1

− · · · ].

So if k is even and k ≥ 8, then for all but ﬁnitely many µ and λ, |N | > |λ|,

and we are done. Similarly, if k is odd and k ≥ 7, then we are done. In the

remaining cases, |N| is given by:

k = 4, |N| = |Rµ − 2λ|

k = 5, |N| = |(Rµ)

− 5Rµλ + 5λ

k = 6, |N| = |(Rµ)

− 4Rµλ + 3λ

One may check that each condition is satisﬁed by only ﬁnitely many rela-

tively prime pairs (µ, λ) with |Rµ −λ| ≤ 2. This concludes the proof in Case

4b.

Case 5. n = 7, and |λ| > 1.

Case 5a. 1/|Rµ − λ| + 1/|λ| < 2/3 and |(Rµ − 2λ)

− 2λ

| > 2|R|.

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 219

We shall consider covers with σ

= id, σ

= σ

−1

, σ

= σ

−1

, and σ

= σ

−1

(see Fig. 9a).

−1

Figure 9. Two covers for n = 7.

We obtain conditions:

[σ

, σ

] = 1(38)

Rµ−λ

= 1(39)

(σ

)

= 1(40)

220 JOSEPH D. MASTERS

Rµ

(σ

−1

)

= 1(41)

(σ

)

Rµ

−2λ

= 1(42)

(σ

)

= 1.(43)

Let us also assume for simplicity that σ

commutes with σ

. Equations (38),

(41) and (42) determine an abelian group A of order |(Rµ − 2λ)

− 2λ

|; we

must show that σ

is non-trivial in A. The elements σ

and σ

generate a

subgroup H of A of index at mos t 2. If σ

= id, then H is cyclic of order

gcd(|λ|, |Rµ − 2λ|). Then |(Rµ − 2λ)

− 2λ

| = |A| ≤ 2|H| = 2gcd(|λ|, |Rµ −

2λ|) = 2gcd(|λ|, |R|) ≤ 2|R|. So if

|(Rµ − 2λ)

− 2λ

| > 2|R|(44)

then σ

6= id. Therefore, under our hypotheses, the relations generate a

group which is isomorphic to the direct sum of a cyclic group with a hy-

perbolic triangle group. As in the previous cases, we may then ﬁnd a non-

peripheral ﬁxed class (in rows 3 and 7), and we are done.

However, note that if R = 1, then Equation (44) is false for all (µ, λ)

satisfying

(µ + 2λ)

− 2λ

= 1.

This is an example of Pell’s equation, which has inﬁnitely many solutions,

and hence (44) may be false inﬁnitely often.

Case 5b. 1/|Rµ − 2λ| + 1/|λ| < 2/3 and |(Rµ − λ)

− 2λ

| > 2|R|.

Let σ

= id, σ

= σ

−1

, σ

= σ

−1

, and σ

= σ

−1

(see Fig. 9b). The

conditions for a cover are:

[σ

, σ

] = 1(45)

Rµ

(σ

−1

)

= 1(46)

(σ

)

Rµ

−λ

= 1(47)

(σ

)

= 1(48)

Rµ−2λ

= 1(49)

(σ

)

= 1.(50)

For simplicity, suppose also that σ

commutes with σ

. Then (45), (46), (47)

determine an abelian group B of order |(Rµ − λ)

− Rµλ|. If σ

= 1, then

|B| ≤ 2gcd(|λ|, |Rµ − λ|) = 2gcd(|λ|, |R|) ≤ 2|R|. Therefore, in this case,

the group determined by conditions (45)-(50) is again the direct pro duct

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 221

of an abelian group with a hyperbolic triangle group, and we may ﬁnd a

non-peripheral ﬁxed class in rows 5 and 7.

Note that Case 5a or 5b applies to all but ﬁnitely many surgeries where

|λ| > 1.

This concludes the proof of Theorem 1.3. 

3. Examples.

We begin with the proof of Theorems 1.5 and 1.6 (see Section 1 for notation).

Lemma 3.1. Let g = D

−1

and h = D

. Then M

−h

(µ, λ)

∼

(µ, λ−

µ), and M

(µ, λ)

∼

(−h)

(µ, λ + µ)

∼

(µ, λ − µ).

Proof. Re call that the mapping class group of the once-punctured torus

is isomorphic to SL

(Z), under the identiﬁcations D

→ R = [

] and

→ L = [

]. Under these identiﬁcations, we compute that h has mon-

odromy matrix [

], (−1) has monodromy matrix [

−1

], and g has mon-

odromy matrix [

−1

−4

]. The homeomorphisms h

and (−h)

have the same

monodromy matrix, and hence are isotopic. Therefore M

∼

(−h)

. Also,

[

−1

−2

](−RL)[

−1

−2

]

−1

= L

−1

, so g and −h have conjugate monodromy

matrices. It follows that M

−h

∼

, and M

(−h)

∼

It remains to determine the eﬀect of these homeomorphisms on the fram-

ings. Computing the maps on π

(F ) gives:

(−h)

]

= (x

−1

yxy

−1

)(h

]

)(x

−1

yxy

−1

)

−1

Therefore the isotopy which takes h

to (−h)

twists ∂F once in a counter-

clockwise manner, so the induced bundle homeomorphism sends M

(µ, λ)

to M

(−h)

(µ, λ + µ).

Let f = D

−1

. The bundle homeomorphism induced by conjugation

preserves the framing, so M

−h

(µ, λ)

∼

f(−h)f

−1

(µ, λ). The homeomor-

phisms f (−h)f

−1

and g have identical monodromy matrices, and hence

are isotopic. We compute g

]

= (yx

−1

x)f(−h)f

−1

]

(yx

−1

so the

isotopy from f (−h)f

−1

to g twists ∂F once in a clockwise manner. The

induced bundle homeomorphism sends M

f(−h)f

−1

(µ, λ) to M

(µ, λ − µ). So

−h

(µ, λ)

∼

(µ, λ − µ).

Likewise, M

f(−h)

−1

(µ, λ)

∼

(µ, λ − 2µ). Thus

(µ, λ)

∼

(−h)

(µ, λ + µ)

∼

f(−h)

−1

(µ, λ + µ)

∼

(µ, λ − µ).



Proof of Theorem 1.5. This is an immediate consequence of Lemma 3.1 and

Theorem 1.3. 

222 JOSEPH D. MASTERS

Proof of Theorem 1.6. We have M(2µ, λ)

∼

(2µ, λ), which is double cov-

ered by M

(µ, λ)

∼

(µ, λ−µ). So it is enough to show that M

(µ, λ−µ)

is virtually Z-representable. By Theorem 1.3, we are done unless

1/| − 2µ − (λ − µ)| + 1/| − 2µ − 2(λ − µ)| + 1/|λ − µ| ≥ 1

or, simplifying:

1/|µ + λ| + 1/|2λ| + 1/|µ − λ| ≥ 1.(51)

By [B3], M(2µ, λ) is virtually Z-representable if 2µ is divisible by 4; hence

we may assume µ is odd. Also, since gcd(2µ, λ) = 1, we may assume λ is

odd, and, assuming (µ, λ) 6= (±1, 1), |λ| 6= |µ|. It follows that

|µ − λ| ≥ 2(52)

|µ + λ| ≥ 2.(53)

The only simultaneous solutions to inequalities (51), (52) and (53) with

µ and λ odd are: (µ, λ) = ±(−3, 1) and ±(3, 1). So the only possible

exceptions to Theorem 1.6 are M(−6, 1)

∼

M(6, 1) and M(2, 1)

∼

M(−2, 1).

The virtual Z-representability of these manifolds may be veriﬁed with either

of the computer programs GAP or Snappea. 

We now turn to the proof of Theorem 1.4.

Proof of Theorem 1.4. Let g and h be as in the statement of Lemma 3.1,

let f = h

, and let i = D

−4

. Both h

and i have monodromy

matrix [

]; hence h

and i are isotopic. By arguments similar to those

used in the proof of Lemma 3.1, we compute that M

(µ, λ)

∼

(µ, µ + λ).

Therefore M

(µ, λ)

∼

(µ, λ + 6µ). Hence by Theorem 1.3 iii, M

(µ, λ)

is virtually Z-representable if

1/|6µ − λ| + 1/|6µ + λ| < 1.(54)

By Lemma 3.1 we have M

(µ, λ)

∼

(µ, λ − 9µ). Hence by Theo-

rem 1.3 ii, M

(µ, λ) is virtually Z-representable if

1/|9µ + λ| + 1/|2λ| + 1/|9µ − λ| < 1.(55)

The only simultaneous solutions to the inequalities 54 and 55 have µ =

0. The proof is completed by noting that M(0, 1) has positive ﬁrst Betti

number, as it is a torus bundle over S

. 

We remark that the same methods may be applied to many other ex-

amples of once-punctured torus bundles, to show that all but ﬁnitely many

surgeries are virtually Z-representable. The idea is to start with a mon-

odromy f to which Theorem 1.3 i or ii applies. Since L

and R generate

a ﬁnite-index subgroup of SL

(Z), there exists an integer ` such that f

VIRTU AL HOMOLOGY OF SURGERED TORUS BUNDLES 223

isotopic to a g satisfying the hypotheses of Theorem 1.3 iii. Usually The-

orem 1.3 will then imply that all but ﬁnitely many surgeries on M

are

virtually Z-representable.

References

[B1] M. Baker, Covers of Dehn ﬁllings on once-punctured torus bundles, Proc. Amer.

Math. Soc., 105 (1989), 747-754.

[B2]

, Covers of Dehn ﬁllings on once-punctured torus bundles II, Proc. Amer.

Math. Soc., 110 (1990), 1099-1108.

[B3] , On coverings of ﬁgure eight-knot surgeries, Paciﬁc J. Math., 150 (1991),

215-228.

[Bart] A. Bart, Surface groups in surgered manifolds, to appear in Topology.

[BZ] S. Boyer and X. Zhang, Virtual Haken 3-manifolds and Dehn ﬁlling, Topology, 39

(2000), 103-114.

[CJR] M. Culler, W. Jaco and H. Rubinstein, Incompressible surfaces in once-punctured

torus bundles, Proc. London Math. Soc., 45(3) (1982), 385-419.

[FH] W. Floyd and A. Hatcher, Incompressible surfaces in punctured torus bundles,

Topology and it Applications, 13 (1982), 263-282.

[H] J. Hempel, Coverings of Dehn ﬁllings of surface bundles, Topology and its Appli-

cations, 24 (1986), 157-170.

[KL] S. Kojima and D. Long, Virtual Betti numbers of some hyperbolic 3-manifolds, A

Fete of Topology, Academic Press, 1988.

[M] S. Morita, Finite coverings of punctured torus bundles and the ﬁrst Betti number,

Sci. Papers College Arts Sci., Univ Tokyo, 35 (1986), 109-121.

[N] A. Nicas, An inﬁnite family of hyperbolic non-Haken 3-manifolds with vanishing

Whitehead groups, Math. Proc. Camb. Phil. Soc., 99 (1986), 239-246.

[V] E.B. Vinberg, Groups deﬁned by periodic paired relations, Sbornik: Mathematics,

188 (1997), 1269-1278.

Received June 15, 1998.

University of Tex as at Austin

Austin, Texas 78712

Rice University

Houston, Texas 77005-1892

E-mail address: mastersj@rice.edu

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

GROUP ACTIONS ON POLYNOMIAL AND POWER

SERIES RINGS

Peter Symonds

When a ﬁnite group G acts faithfully on a graded integral

domain S which is an algebra over a ﬁeld k, such as a poly-

nomial ring, we consider S as a kG-module. We show that S

is asymptotically mostly projective in each degree, and also

that it is in fact mostly free in an appropriate sense. Simi-

lar results also hold for ﬁltered algebras, such as power series

rings.

1. Introduction.

Let S =

∞

n=0

be a graded algebra over a ﬁeld k. We suppose that

S is ﬁnitely generated over k as a k-algebra and that the homogeneous

components S

are ﬁnite dimensional vector spaces over k. Let G be a ﬁnite

group of grading preserving automorphisms of S (so G acts faithfully). We

are concerned with the structure of S as a kG-module.

The classical theory of Hilbert and Serre asserts that for large n, dim

is given by a function

(n) = c

d−1

(n)n

d−1

+ c

d−2

(n)n

d−2

+ · · · + c

(n),

where the c

(n) are rational valued functions periodic in n, i.e., φ

(n +

p) = φ

(n) for some integer p (see Section 2). If c

d−1

is assumed not to be

identically zero then d is equal to the dimension of the ring in various senses.

If S is a polynomial ring then d is equal to the number of variables.

From now on, we assume that S is an integral domain. Let P

denote the

maximal projective summand of S

(deﬁned up to isomorphism).

Theorem 1.1. dim

) is bounded by a polynomial in n of degree d−2.

Thus S

is mostly projective, and if S is a polynomial ring then the

non-projective part grows like a polynomial ring in one fewer variables.

In fact S is mostly free, although the individual S

do not have to contain

a free module at all; the diﬀerent projectives can occur in diﬀerent degrees.

To explain this let R = S

, the ring of invariants.

225

226 PETER SYMONDS

Theorem 1.2. S contains a free kG-submodule F of rank 1, a sum of ho-

mogeneous pieces, such that the product map R ⊗

F → S is injective. De-

note its image by RF =

(RF )

. Then RF is a free summand of S and

dim

/(RF )

) is bounded by a polynomial of degree d − 2.

Of course, the ﬁrst theorem is a corollary to the second. Versions of these

theorems were proven by Howe [4] in characteristic 0 and by Bryant [2, 3]

for polynomial rings.

Section 2 contains the main proof, except for some technical details which

appear in Section 3. Section 4 proves similar results for ﬁltered algebras.

2. Main Proof.

Proof. We can assume that k is a splitting ﬁeld for G, since if a kG-module

contains a free or projective summand after extension of scalars then it did

so before. Let Q

(resp. Q

) denote the ﬁelds of fractions of S (resp. R),

so Q

∼

⊗

S. By the Normal Basis Theorem, Q

is a free module of

rank 1 over Q

G; let e be a generator. Then, over kG, e generates a free

submodule E of rank 1 such that Q

∼

⊗

E. Now there is an r ∈ R

such that re ∈ S. Let F be the kG-module generated by re, so F ⊂ S and

F = rE

∼

kG. Also the product map R ⊗

F → RF ⊂ S is injective.

We claim that F can be assumed to be a sum of homogeneous pieces,

F =

. The proof of this plausible statement is somewhat delicate,

and we postpone it to the next section.

Let x

, . . . , x

be homogeneous generators for S as a k-algebra. Then

, where a

i,j

, b

i,j

∈ R and the e

form a homogeneous k-basis

for F . By writing b

i,j

and taking the homogeneous component

of this equation in some degree where b

i,j

is non-zero, we see that we may

assume that the b

i,j

are homogeneous. Let α ∈ R

be the product of all the

i,j

. Then each x

∈ α

−1

RF , so S ⊂ α

−1

RF .

Thus

(RF )

⊂ S

⊂ α

−1

(RF )

n+a

and so, identifying RF with R ⊗

F , we have

n−n

⊗ F

⊂ S

⊂ α

−1

n+a−n

⊗ F

In particular, the dimension of S

/(RF )

is bounded by the diﬀerence in

the dimensions of the two sides, i.e., by

(φ

(n + a − n

) − φ

(n − n

)) dim

GROUP ACTIONS ON POLYNOMIAL RINGS 227

But

(n + a − n

) − φ

(n − n

) = c

d−1

(n + a − n

)(n + a − n

)

d−1

− c

d−1

(n − n

)(n − n

)

d−1

+ lower degree terms,

and c

d−1

is periodic, with period dividing a (see 3.1), so the n

d−1

term

cancels and we are done. 

3. Technical Details.

The form of φ

(n) given above is not quite the standard one, although it is

quoted in [4]. The usual references deal with a module over a polynomial

ring which has all the variables in degree 1, and then all the coeﬃcients of

φ are constants. To deduce the version given in the introduction, note that

if S is generated by x

, . . . , x

then it is a ﬁnitely generated module over

k[x

, . . . , x

]. By taking suitable powers y

of the x

we can get all the y

the same degree b, and S will still be ﬁnitely generated over k[y

, . . . , y

For 0 ≤ j ≤ b − 1, set T

∞

l=0

j+lb

. Then R

∼

as a k[y

, . . . , y

module, and after regrading each T

so that each y

can have degree 1, we

can apply the usual theory ([1] 11.2, [5] VII Theorem 41) to each T

and

sum the results. It is the summation that leads to the periodic coeﬃcients.

Lemma 3.1 ([4]). If R is an integral domain (as it always is for us), c =

gcd{r ∈ Z|R

6= 0} and φ

(n) = c

d−1

(n)n

d−1

+ · · · + c

(n), then there is a

constant b such that

d−1

(n) =



b, if n|c,

0, otherwise.

Proof. If 0 6= α ∈ R

, then multiplication by α embeds R

in R

n+a

, so

for large n, φ

(n) ≤ φ

(n + a). Now consider the limit of φ

(n)/n

d−1

n → ∞ through elements of the same residue class modulo the period of

d−1

to see that c

d−1

(n) ≤ c

d−1

(n + a). This, together w ith the periodicity,

implies the result. 

Now we prove the claim made in Section 2.

Proposition 3.2. The free module F ⊂ S can be assumed to be a sum of

homogeneous pieces in such a way that the product map Q

⊗

F → Q

still an isomorphism.

Proof. For each simple kG-module V , let T

= Hom

(V, S), a graded R-

module. Now soc F is a direct sum of simples. Let soc

(F ) denote the sum

of those isomorphic to V , so soc

(F ) = V

⊕· · ·⊕V

, where V

∼

V , and let

be a projective summand of F with soc(P

) = V

. The inclusions of the

in S give us s homomorphisms f

∈ T

, which are linearly independent

over R.

228 PETER SYMONDS

Lemma 3.3. Let f

, . . . , f

be elements of a graded R-module T which are

linearly independent over R. Write each f

as a sum of its homogeneous

components; f

, f

∈ T

. Then for each j there is an integer k

such that f

, . . . , f

are linearly independent over R.

Proof. For each 0 ≤ t ≤ s, let P

be the claim that there exist integers

, . . . , k

such that f

, . . . , f

, f

t+1

, . . . , f

are linearly independent over

R. P

is true by hypothesis and we want P

. We give a proof by induction

on t, so assume P

If P

t+1

is false, then for each k ∈ Z we can ﬁnd u

, r

∈ R, u

6= 0, such

that

t+1

= r

+ · · · + r

+ r

t+2

+ · · · + r

Let u be the product of the u

for which f

t+1

6= 0. Then uf

t+1

(

t+1

, contradicting P

. 

Applying this to the {f

} ⊂ T

we obtain homogeneous {

} ⊂ T

∈

, say, linearly independent over R.

Lemma 3.4. The evaluation map ev : T

⊗

V → S is injective.

Proof. In fact ev : Hom

(V, M) ⊗

V → M is injective for any kG-module

M. This is because it factors through soc

(M), which is a direct sum

of V ’s, so we are reduced to proving the case M = V . But then ev is an

isomorphism, since Hom

(V, V )

∼

k, by the assumption that k is a splitting

ﬁeld. 

Corollary. The product map R ⊗

(

(V )) → S is injective.

Now let

be the image of the projection of P

to S

. The projection

map is injective on soc(P

), by the construction of a

, so

∼

and

soc(

) =

(V ). Let

and consider the product map R ⊗

→ S. Since soc(R ⊗

) = R ⊗

(V ), it is injective on the socle,

so is injective.

Finally, we sum the

over the simples V to obtain

F , a free kG-module

of rank 1, which is a sum of homogeneous pieces, as required. 

Remark. If G is a p-group, where p is the characteristic of k, then the proof

is much simpler because soc(F )

∼

k. Under at least one of the projections

of F onto its homoge neous components the image of soc(F ) must be non-

zero. Let

F be the image of F under this projec tion. Then

∼

kG and

⊗

F → Q

is an isomorphism because it is injective on the socle, and

both sides have the same dimension over Q

This is enough to prove 1.1 for general G. For if P = Syl

(G) then S is a

direct summand of Ind

Res

GROUP ACTIONS ON POLYNOMIAL RINGS 229

Remark. It is not hard to see that, given any degree m, the summ ands of

F can be moved by multiplication by a scalar to lie in T

m+lc

= S

m+lc

⊕· · · ⊕

m+(l+1)c−1

, for some l. The argument of the proof of 1.2 now shows that

the non-free part of T

has dimension bounded by a polynomial of degree

d − 2 (cf. [2, 3]).

4. Filtered Rings.

The case of ﬁltered rings is slightly diﬀerent. Consider the power series ring

k[[x]] in characteristic 2 and let the group of order 2 act by x 7→ x/(x + 1) =

x + x

+ x

+ · · · . The action on the associated graded ring is trivial, yet

the action on k[[x]] certainly c ontains free summands (the only alternative

is trivial).

We consider ﬁnitely generated k-algebras S which are integral domains

and have a ﬁltration S = I

⊂ I

⊂ · · · . Each S/I

is assumed to

be ﬁnite dimensional over k, and ∩I

= {0}. There is a ﬁnite group G

of automorphisms of S, which preserves the ﬁltration. The invariants are

R = S

with the induced ﬁltration J

= R ∩ I

. Again there is a function

(n) = c

(n)n

+ c

d−1

+ · · · c

(n),

where the c

are periodic, such that dim

(S/I

) = χ

(n) for large n. If S is

a power series ring, then d is equal to the number of variables.

As before there is a free kG-module of rank 1 in S, and the product map

R ⊗

F → S is injective. Since F is ﬁnite dimensional there is s ome integer

f such that F ∩ I

= 0, so F injects into S/I

For each n, let K

be a vector space complement to J

in R. Then the

product map K

⊗ F → S/I

f+n

is injective, so its image, K

F is a free

summand of S/I

f+n

Proceeding in the same way as before we can prove:

Theorem 4.1. dim

((S/I

f+n

)/K

F ) is bounded by a polynomial of degree

d − 1.

So S is mos tly free. Again, for a power series ring, the non-free part grows

like a power series ring in one fewer variables.

References

[1] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-

Wesley, Reading, MA, 1969.

[2] R.M. Bryant, Symmetric powers of representation of ﬁnite groups, Jour. Algebra, 154

(1993), 416-436.

[3] , Groups acting on polynomial algebras, Finite and locally ﬁnite groups. NATO

Advanced Science Institutes Series C: Mathematical and Physical Sciences, 471, 327-

346, Kluwer Acad. Publ. Dordrecht, 1995.

230 PETER SYMONDS

[4] R. Howe, Asymptotics of dimensions of invariants for ﬁnite groups, Jour. Algebra,

122 (1989), 374-379.

[5] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Van Nostrand, Princeton,

1960.

Received October 13, 1998.

Department of Mathematics

U.M.I.S.T.

Manchester M60 1QD

England

E-mail address: psymonds@umist.ac.uk

PACIFIC JOURNAL OF MATHEMATICS

Vol. 195, No. 1, 2000

THE MODULI OF FLAT PU(2,1) STRUCTURES ON

RIEMANN SURFACES

Eugene Z. Xia

For a compact Riemann surface X of genus g > 1,

Hom(π

(X) , PU(p, q))/PU(p, q) is the moduli space of ﬂat

PU(p, q)-connections on X. There are two integer invari-

ants, d

, d

, associated with each σ ∈ Hom(π

(X) , PU(p, q))/

PU(p, q). These invariants are related to the Toledo invari-

ant τ by τ = 2

−pd

p+q

. This paper shows, via the theory of

Higgs bundles, that if q = 1, t hen −2(g − 1) ≤ τ ≤ 2(g − 1).

Moreover, Hom(π

(X) , PU(2, 1))/PU(2, 1) has one connected

component corresponding to each τ ∈

Z with −2(g − 1) ≤

τ ≤ 2(g − 1). Therefore the total number of connected com-

ponents is 6(g − 1) + 1.

1. Introduction.

Let X be a smooth projective curve over C with genus g > 1. The deforma-

tion space

= Hom

(π

(X), PGL(n, C))/ PGL(n, C)

is the space of equivalence classes of semi-simple PGL(n, C)-represe ntations

of the fundamental group π

(X). This is the PGL(n, C)-Betti moduli space

on X [22, 23, 24]. A theorem of Corlette, Donaldson, Hitchin and Simpson

relates CN

to two other moduli spaces, CN

and CN

Dol

—the PGL(n, C)-

de Rham and the PGL(n, C)-Dolbeault moduli spaces, respectively [3, 5, 11,

21]. The Dolbeault moduli space consists of holomorphic objects (Higgs

bundles) over X; therefore, the classical results of analytic and algebraic

geometry can be applied to the study of the Dolbeault mo duli space.

Since PU(p, q) ⊂ PGL(n, C), CN

contains the space

= Hom

(π

(X), PU(p, q))/ PU(p, q).

The space N

will be referred to as the PU(p, q)-Betti moduli space which

similarly corresponds to some subspaces N

and N

Dol

of CN

and CN

Dol

respectively. We shall refer to N

and N

Dol

as the PU(p, q)-de Rham and

the PU(p, q)-Dolbe ault moduli spaces.

The Betti moduli spaces are of gre at interest in the ﬁeld of geometric

topology and uniformization. In the case of p = q = 1, Goldman analyzed

231

232 EUGENE Z. XIA

and determined the number of its connected c omponents to be 4g − 3

[6]. Hitchin subsequently considered N

Dol

in the case of p = q = 1 and

determined its topology [11].

In this paper, we analyze N

Dol

for the case of p = 2, q = 1 and deter-

mine its number of connected components. In addition, we produce a new

algebraic proof, via the Higgs-bundle theory, of a theorem by Toledo on the

bounds of the Toledo invariant [26, 27].

An element σ ∈ Hom

(π

(X), PU(p, q)) deﬁnes a ﬂat principal PU(p, q)-

bundle P over X. Such a ﬂat bundle may be lifted to a principal U(p, q)-

bundle

P with a Yang-Mills connection D [2, 3, 5, 11, 21]. Let E be the

rank-(p + q) vector bundle associated with (

P , D). The second c ohomology

(X, Z) is isomorphic to Z, so one may identify the Chern class c

(E) ∈

(X, Z) with an integer, the degree of E. Suppose we impose the additional

condition

0 ≤ deg(E) < n.

Then the above construction give s rise to a unique obstruction class o

(E) ∈

(X, π

(U(p, q))) [25]. The obstruction class is invariant under the conju-

gation action of PU(p, q); therefore, one obtains the obstruction map:

: Hom

(π

(X), PU(p, q))/ PU(p, q) −→ H

(X, π

(U(p, q)))

∼

Z × Z.

The maximum compact subgroup of U(p, q) is U(p) × U(q). Hence topo-

logically E is a direct sum E

⊕ E

with

deg(E) = deg(E

) + deg(E

The obstruction class o

(E) is then (deg(E

), deg(E

)) ∈ Z ×Z. Associated

with σ is the Toledo invariant τ which relates to d

= deg(E

) and d

deg(E

) by the formula [7, 26, 27]

τ = 2

deg(E

⊗ E

∗

)

p + q

= 2

− pd

p + q

This explains why the Toledo invariant of a PU(2, 1) representation cannot

be an odd integer [7]. The main result presented here is the following:

Theorem 1.1. Hom

(π

(X), PU(2, 1))/ PU(2, 1) has one connected com-

ponent for each τ ∈

Z with −2(g − 1) ≤ τ ≤ 2(g − 1). Therefore the total

number of connected components is 6(g − 1) + 1.

We shall also provide a new proof en route to the following theorem:

Theorem 1.2 (Toledo). Suppose σ ∈ Hom

(π

(X), PU(p, 1)) and τ is the

Toledo class of σ. Then

−2(g − 1) ≤ τ ≤ 2(g − 1).

Moreover τ = ±2(g − 1) implies σ is reducible.

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 233

These results are related to the results of Domic and Toledo [4, 26, 27]

and, as being pointed out to the author recently, are also related to the work

of Gothen [8] which computed the Poincar´e polynomials for the components

of Hom(π

(X), PSL(3, C))/ PSL(3, C), where deg(E) is coprime to 3.

Acknowledgments.

Most of this research was c arried out while the author was at the Univer-

sity of Maryland at College Park. I thank J. Adams, K. Coombes, P. Green,

K. Joshi, S. Kudla, P. Newstead, J. Poritz and especially W. Goldman and

C. Simpson for insightful discussions over the course of the research. I thank

the referee for helpful suggestions.

2. Backgrounds and Preliminaries.

In this section, we brieﬂy outline the constructions of the Betti, de Rham

and Dolbeault moduli spaces. For details, see [2, 3, 5, 11, 12, 18, 21, 22,

23, 24].

2.1. The Betti Moduli Space. The fundamental group π

(X) is gener-

ated by S = {A

, B

}

i=1

, subject to the relation

i=1

−1

= e.

Denote by I and [I] the identities of GL(n, C) and PGL(n, C), respectively.

Deﬁne

R : PGL(n, C)

−→ PGL(n, C)

R : GL(n, C)

−→ GL(n, C)

to be the commutator maps:

, Y

, . . . , X

, Y

)

R,R

−→

i=1

−1

The group

{ζI : ζ ∈ C, ζ

= 1}

is isomorphic to Z

. The space R

−1

) is identiﬁed with the representation

space Hom(Γ, GL(n, C)), where Γ is the central extension [2, 11]:

0 −→ Z

−→ Γ −→ π

(X) −→ 0.

Each element ρ ∈ R

−1

) acts on C

via the standard representation of

GL(n, C). The representation ρ is called reducible (irreducible) if its action

on C

is reducible (irreducible). A representation ρ is called semi-simple if

it is a direct sum of irreducible representations. Let ζ

= e

2πi/n

and deﬁne

−1

(ζ

I) : σ is semi-simple}/ GL(n, C),

234 EUGENE Z. XIA

n−1

[

c=0

(c),

(c)/ Hom(π

(X), C

∗

)

= Hom

(π

(X), PGL(n, C))/ PGL(n, C).

Fix p, q such that p + q = n. Denote by R

the restriction of R to the

subgroup U(p, q)

. Deﬁne

−1

(ζ

I) : σ is semi-simple}/ U(p, q),

n−1

[

c=0

(c).

Note the center of U(p, q) is U(1) and is contained in the center of GL(n, C).

It follows that M

(c). Deﬁne

(c)/ Hom(π

(X), U(1))

= M

/ Hom( π

(X), U(1)) = Hom

(π

(X), U(p, q))/ U(p, q).

All the spaces constructed here that contain the symbols M

or N

will

be loosely referred to as Betti moduli spaces. The subspace of irreducible

elements of a Betti moduli space will be denoted by an s superscript. For

example, CM

denotes the subspace of irreducible elements of CM

2.2. The de Rham Moduli Space. Suppose P is a principal GL(n, C)-

bundle on X, E its associated vector bundle of rank n and G

(E) the group

of GL(n, C)-gauge transformations on E. A connection is called Yang-Mills

(or central) if its curvature is c entral [2]. The gauge group G

(E) acts

on the space of GL(n, C)-connections on E and preserves the subspace of

Yang-Mills connections. Fix E with deg(E) = c. The de Rham moduli space

(E)-equivalence classes of Yang-Mills

connections.

Let M

U(p, q)-central connections on E. In other words, M

(c), but with U(p, q) replacing GL(n, C). Since the center of U(p, q)

is contained in the center of GL(n, C), M

(c).

The space of C

∗

-gauge equivalence classes of C

∗

-connections on X is

(X, C

∗

) which acts on CM

(c).

This action corresponds to the action of Hom(π

(X), C

∗

) on CM

the quotient CN

(c). Similarly, the space of U(1)-

gauge equivalence classes of U(1)-connections on X is H

(X, U(1)) which

acts on M

(c). De ﬁne

∞

[

c=−∞

(c), CN

∞

[

c=−∞

(c)

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 235

∞

[

c=−∞

(c), N

∞

[

c=−∞

(c).

All the spaces constructed here that contain the symbols M

or N

will be loosely referred to as de Rham mo duli spaces. A central connection

is irreducible if (E, D) = (E

⊕ E

, D

⊕ D

) implies rank(E

) = 0 or

rank(E

) = 0. The subspace of irreducible elements of a de Rham moduli

space will be denoted by an s superscript.

Theorem 2.1. The moduli space CM

(c).

Proof. See [3, 5, 11]. 

Consider all the objects we have deﬁned so far with subscripts B or DR.

With Theorem 2.1, one can verify the following: Suppose two objects have

subscripts B or DR. Then the two objects are homeomorphic if they only

diﬀer in subscripts. For example, N

(c).

Since the maximum compact subgroup of U(p, q) is U(p) × U(q), (E, D) ∈

implies E is a direct sum of a U(p) and a U(q)-bundle:

E = E

⊕ E

where the ranks of E

and E

are p and q, respectively. Therefore, associated

to each (E, D) are the invariants

= deg(E

) and d

= deg(E

with

+ d

= deg(E) = c.

The Toledo invariant τ is [7, 26, 27]

τ = 2

deg(E

⊗ E

∗

)

= 2

− pd

The subspace of M

By the equivalence of Betti and de Rham moduli spaces, one may deﬁne the

Toledo invariant on M

(c). Denote by M

the subspace of M

a ﬁxed Toledo invariant τ. The H

(X, U(1)) action on M

and the quotient is denoted by N

. In the Betti moduli space, the

Hom(π

(X), U(1)) action on M

preserves M

, and the quotient is denoted

by N

2.3. The Dolbeault Moduli Space. Let E be a rank n complex vector

bundle over X with deg(E) = c. Denote by Ω the canonical bundle on

X. A holomorphic structure ∂ on E induces holomorphic structures on the

bundles End(E) and End(E) ⊗ Ω. A Higgs bundle is a pair (E

∂

, Φ), where

∂ is a holomorphic structure on E and Φ ∈ H

(X, End(E

∂

) ⊗ Ω). Such a Φ

is called a Higgs ﬁeld. We denote the holomorphic bundle E

∂

by V .

236 EUGENE Z. XIA

Deﬁne the slope of a Higgs bundle (V, Φ) to be

s(V ) = deg(V )/ rank(V ).

For a ﬁxed Φ, a holomorphic subbundle W ⊂ V is said to be Φ-invariant if

Φ(W ) ⊂ W ⊗Ω. A pair (V, Φ) is stable (semi-stable) if W ⊂ V is Φ-invariant

implies

s(W ) < (≤)s(V ).

A Higgs bundle is called poly-stable if it is a direct sum of stable Higgs

bundles of the same slope [11, 22].

The gauge group G

(E) acts on holomorphic structures by pull-back and

on Higgs ﬁelds by conjugation. Moreover the G

(E) action preserves stabil-

ity, poly-stability and semi-stability. The Dolbeault moduli space CM

Dol

(c)

on E (with deg(E) = c), is the G

(E)-equivalence classes of poly-stable

(or S-equivalence classes of semi-stable [18]) Higgs bundles (V, Φ) on X

[11, 12, 18, 22]. A Higgs bundle is called reducible if it is poly-stable but

not stable. Let

Dol

∞

[

c=−∞

Dol

(c).

If D ∈ CM

(c), then for any Hermitian metric h on E, there is a

decomposition,

D = D

+ Ψ,

where D

is compatible with h and Ψ is a 1-form with coeﬃcients in p. The

(0, 1) part of D

determines a holomorphic structure ∂

on E while the

(1, 0) part of Ψ is a section of the bundle End(E) ⊗ Ω. There exists a metric

h such that the pair

(V, Φ) = (E

∂

, Ψ

1,0

)

so constructed is a poly-stable Higgs bundle [11, 21, 22]. Therefore this

construction gives a map

f : CM

Dol

(c).

Theorem 2.2 (Corlette, Donaldson, Hitchin, Simpson). The map f is a

homeomorphism.

Proof. See [3, 5, 11, 21]. 

3. The U(p, q)-Yang-Mills Connections.

Assume p ≥ q and p + q = n. From the previous section, we know that

⊂ CM

. Let D ∈ CM

on a rank n vector bundle

E −→ X.

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 237

Proposition 3.1. D is a U(p, q)-Yang-Mills connection if and only if its

corresponding Higgs bundle (V, Φ) ∈ CM

Dol

conditions:

1) V is decomposable into a direct sum:

V = V

⊕ V

where V

, V

are of rank p, q, respectively.

2) The Higgs ﬁeld decomposes into two maps:

: V

−→ V

⊗ Ω,

: V

−→ V

⊗ Ω.

Proof. Suppose D is a U(p, q)-Yang-Mills connection. Denote by h the

Hermitian-Yang-Mills metric on (E, D). Then D decomposes as

D = D

+ Ψ,

where D

is the part compatible with h. The Cartan decomposition (g =

k ⊕ p) for u(p, q) is

u(p, q) = (u(p) ⊕ u(q)) ⊕ p.

If we take the standard representation of u(p, q), then elements in k are of

the form



a 0

0 d



where a ∈ u(p), b ∈ u(q), respectively. The elements in p are then of the

form



0 b

c 0



where b ∈ Hom(C

, C

), c ∈ Hom(C

, C

), respectively. Hence on local

charts, D

and Ψ have coeﬃcients in k and p, respectively. In particular,

the connection D

is reducible.

The Higgs bundle corresponding to D is (E

∂

, Φ) where ∂

is the (0, 1)-

part of D

and Φ, the (1, 0)-part of Ψ, is considered as a holomorphic bundle

map:

Φ : V −→ V ⊗ Ω.

Since D

has coeﬃcient in k, the holomorphic structure on V deﬁned by

0,1

is a direct sum:

V = V

⊕ V

Since Ψ is block oﬀ-diagonal, Φ is also block oﬀ-diagonal implying Φ can be

decomposed into two maps:

: V

−→ V

⊗ Ω,

: V

−→ V

⊗ Ω.

This proves the only if part of the proposition.

238 EUGENE Z. XIA

Suppose (V, Φ) is a Higgs bundle that satisﬁes the two conditions of Propo-

sition 3.1. Let α be the constant gauge

α =



0 −I



where I

, I

are p × p, q × q identity matrices, respectively. Then α acts on

the space of holomorphic structures on E and ﬁxes V . Moreover,

αΦα

−1

= −Φ

since Φ is of the form

Φ =



0 Φ



Hence by a theorem of Simpson, the corresponding Hermitian-Yang-Mills

metric h is invariant under the action of α [21]. In other words, on local

charts, h is a Hermitian matrix of the form

h =



a 0

0 d



where a, d are Hermitian matrices of dimension p × p, q × q, respectively.

Hence the corresponding Yang-Mills connection is

D = D

+ Φ + Φ

‡

where Φ

‡

is the adjoint of Φ with respect to h. In local coordinates, D

has

coeﬃcient of the form



a 0

0 d



and Φ + Φ

‡

is of the form



0 b

‡



Hence D

and Φ + Φ

‡

have coeﬃcients in u(p) ⊕ u(q) and p, respectively.

This implies D is a U(p, q)-Yang-Mills connection. 

Denote by M

Dol

Proposition 3.1. Then M

Dol

(c).

The invariants d

, d

and τ on (E, D) translate to invariants on the

corresponding U(p, q)-Higgs bundles (V

⊕ V

, Φ):

= deg(V

), d

= deg(V

), τ = 2

− pd

The subspace of M

Dol

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 239

4. Group Actions and K¨ahler Structures on CM

Dol

4.1. The Action of line bundles. The space of holomorphic line bundles,

(X, O

∗

), acts freely on CM

Dol

as follows:

(X, O

∗

) × CM

Dol

7−→ CM

Dol

(L, (V, Φ)) 7−→ (V ⊗ L, Φ ⊗ 1),

where 1 is the identity map on L. An immediate consequence is:

Proposition 4.1. If c

≡ c

mod n, then CM

Dol

) is homeomorphic to

Dol

4.2. The Action of H

(X, Ω). The vector space H

(X, Ω) acts freely on

Dol

as follows:

(X, Ω) × CM

Dol

7−→ CM

Dol

(φ, (V, Φ)) 7−→ (V, Φ + φI).

The actions of H

(X, O

∗

) and H

(X, Ω) commute and the quotient is

deﬁned to be

Dol

= CM

Dol

/(H

(X, O

∗

) × H

(X, Ω)).

The H

(X, O

∗

) action preserves the subspaces M

Dol

. The

quotients are deﬁned to be

Dol

(c)/H

(X, O

∗

Dol

= M

Dol

(X, O

∗

All the spaces constructed so far that contain the symbols M

Dol

or N

Dol

will be loosely referred to as the Dolbeault moduli spaces. The subspace

of stable Higgs bundles of a Dolbeault moduli space will be denoted by an

s superscript. For example, CM

Dol

will denote the subspace of irreducible

elements of CM

Dol

Remark 1. The Betti, de Rham and Dolbeault moduli spaces CM

Dol

and CM

Dol

constructed here are variations of those of Simpson’s

[22, 23, 24].

With Theorems 2.1 and 2.2, one can obtain the following equivalence

relations between the various Betti, de Rham and Dolbeault moduli spaces.

Corollary 4.2. Suppose M

⊂ M

(c). Then one obtains the following

commutative diagram:

−→ M

(c)



−→ M

(c)



Dol

−→ M

Dol

(c).

240 EUGENE Z. XIA

Moreover the horizontal maps are continuous injections and vertical maps

are homeomorphisms. One obtains three additional commutative diagrams

by respectively replacing the symbol M by M

, N and N

in the above

diagram. In the case of M

, the maps in the commutative diagram are

smooth.

4.3. The Dual Higgs Bundles. There is a Z

action on CM

Dol

. Let

(V, Φ) ∈ CM

Dol

where Φ is a holomorphic map:

Φ : V −→ V ⊗ Ω.

This induces a map on the dual bundles

∗

: V

∗

⊗ Ω

∗

−→ V

∗

Tensoring with Ω,

∗

⊗ 1 : V

∗

−→ V

∗

⊗ Ω,

where 1 denotes the identity map on Ω. This produces the dual Higgs bundle

∗

, Φ

∗

⊗ 1). We shall abbreviate it as (V

∗

, Φ

∗

Proposition 4.3. If (V, Φ) ∈ CM

Dol

(c), then (V

∗

, Φ

∗

) ∈ CM

Dol

(−c).

Proof. One must show that (V, Φ) is stable (semi-stable) implies (V

∗

, Φ

∗

) is

stable (semi-stable). Suppose W

⊂ V

∗

is Φ

∗

-invariant. Then we have the

following commutative diagram

0 −→ W

−→ V

∗

−→ W

−→ 0



∗



∗



∗

0 −→ W

⊗ Ω −→ V

∗

⊗ Ω −→ W

⊗ Ω −→ 0

where W

= V

∗

. The proposition follows by dualizing the diagram. 

In light of Propositions 4.1 and 4.3 we have:

Corollary 4.4. If c

= ±c

mod n, then CM

Dol

) is homeomorphic to

Dol

4.4. The U(1) and C

∗

-Actions on the Complex Moduli Spaces. If

(V, Φ) ∈ CM

Dol

(c), then for t ∈ C

∗

, (V, tΦ) ∈ CM

Dol

(c). This deﬁnes an

analytic action [11, 12, 22]

∗

× CM

Dol

(c).

Since U(1) ⊂ C

∗

, this also induces a U(1)-action on CM

Dol

(c).

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 241

4.5. The Moment Map. The moduli space CM

Dol

(c)

is K¨ahler [11, 12].

Denote by i, ω the corresponding complex and symplectic structures, respec-

tively. Deﬁne the Morse function [11, 12]

m : CM

Dol

(c)

−→ R,

m(V, Φ) = 2i

tr(ΦΦ

‡

where Φ

‡

is the adjoint of Φ with respect to the Hermitian-Yang-Mills metric

on (E, D). Denote by X the vector ﬁeld on CM

Dol

(c)

such that [12]

grad m = iX.

Theorem 4.5.

1) The map m is proper.

2) The U(1)-action generates X.

3) The C

∗

action is analytic with respect to i; therefore, the orbit of C

∗

is locally an analytic subvariety with respect to i.

Proof. See [11, 12, 22]. 

Corollary 4.6. Each component of CM

Dol

local minimum of m.

Corollary 4.7. If the C

∗

action preserves M ⊂ CM

Dol

(c)

, then the gra-

dient ﬂow grad m preserves M.

Let m

be the restriction of m to the subspace M

Dol

⊂ CM

Dol

(c).

Corollary 4.8. Every component of M

Dol

contains a point that is a local

minimum of m

. If (V, Φ) is stable and is a local minimum of m

, then

(V, Φ) is a critical point of m.

Proof. Consider

⊂ M

(c).

Since U(p, q) is closed in GL(n, C), M

(c).

Since the obstruction map o

is continuous, M

is a closed subspace of

(c). Hence M

is closed in CM

(c). Hence by Theorem 4.5, m

proper. Thus each c omponent of M

Dol

contains a local minimum of m

The points in (M

Dol

)

are smooth. Suppose (V, Φ) ∈ (M

Dol

)

. Then

(V, Φ) is of the form described in Propos ition 3.1. Hence the C

∗

action

preserves the subspace (M

Dol

)

⊂ CM

Dol

. By Corollary 4.7, the gradient

ﬂow of m preserves (M

Dol

)

. Hence

grad m

= grad m = iX.

If m

is a local minimum at (V, Φ), then

grad m(V, Φ) = grad m

(V, Φ) = 0.

Hence (V, Φ) is a c ritical point of m. 

242 EUGENE Z. XIA

5. Bounds on Invariants.

In this section, we assume q = 1 and let n = p + q = p + 1. In light of

Proposition 4.3 and Corollary 4.4, one may further as sume that τ ≥ 0 and

0 ≤ c < n, or equivalently,

s(V

) ≤ s(V ) ≤ s(V

), 0 ≤ c < n.

Proposition 5.1. If (V, Φ) = (V

⊕ V

, (Φ

, Φ

)) ∈ M

Dol

(c)

Dol

(c)),

then

< (≤)

c(n − 1)

+ (g − 1)

> (≥)

− (g − 1).

Proof. Suppose (V

⊕ V

, Φ) ∈ M

Dol

(c)

with Φ = (Φ

, Φ

) in the notation

of Proposition 3.1. Since s(V

) ≥ s(V ),

: V

−→ V

⊗ Ω

is non-zero.

Construct the canonical factorization for Φ

[20]: There exist holomorphic

bundles V

, V

and W

, W

such that the following diagram

0 −→ V

−→ V

−→ 0



0 ←− W

←− V

⊗ Ω

←− W

←− 0

commutes, and the rows are exact, rank(V

) = rank(W

) and ϕ has full rank

at a generic point of X. This implies

(

deg(V

) + deg(V

) = d

deg(W

) + deg(W

) = d

+ 2(g − 1).

Since Φ

6≡ 0, we have ϕ 6≡ 0, rank(W

) = 0 and W

= V

⊗ Ω.

The case of p = 1 has been dealt with by Hitchin [11], so we assume p > 1.

Then V

is a Φ-invariant subbundle of positive rank. Stability implies

s(V

) < s(V ) = (d

+ d

)/n = c/n.

Since the map

−→ W

= (V

⊗ Ω)

is not trivial,

deg(V

) ≤ deg(W

) = deg(V

⊗ Ω).

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 243

So one has











s(V

) < s(V )

= deg(V

) + deg(V

)

deg(V

) ≤ d

+ 2(g − 1).

This implies

(n − 2)c

+ d

+ 2(g − 1).

Since d

+ d

= c,

c(n − 1)

+ (g − 1)

and

− (g − 1).

When (V, Φ) is semi-stable, one has either Φ 6≡ 0 or Φ ≡ 0. In the former

case, one has s(V

) ≤ s(V ) implying

≤

c(n − 1)

+ (g − 1)

≥

− (g − 1).

In the latter case , V

is Φ-invariant. By the assumption s(V

) ≤ s(V

= d

= 0 and τ = 0. 

By deﬁnition,

τ = 2

− pd

≤



c(n − 1)

+ (g − 1) − (n − 1)

+ (n − 1)(g − 1)



= 2(g − 1).

Equality holds only when (V, Φ) is semi-stable but not stable, in which case,

the associated ﬂat connection is reducible. This proves Theorem 1.2.

6. Reducible Higgs Bundles.

Let p = 2 and q = 1 and assume τ ≥ 0 and 0 ≤ c < 3. By deﬁnition, a

reducible poly-stable Higgs bundle is a direct sum of stable Higgs bundles

of the same slope. These Higgs bundles correspond to the reducible repre-

sentations in M

. A direct computation shows that if (V, Φ) is reducible,

then

deg(V ) = d

+ d

= 0

and the associated Toledo invariant τ is an even integer. Hence one has:

244 EUGENE Z. XIA

Proposition 6.1. If c = deg(V ) 6= 0 and (V, Φ) ∈ M

Dol

(c), then (V, Φ) is

stable. In particular, M

Dol

An example of a reducible Higgs bundle is (O ⊕ Ω

⊕ Ω

−

, Φ), where

Φ : Ω

−→ Ω

−

⊗ Ω

is a holomorphic bundle isomorphism. That is, Φ is of the form





0 0 0

0 1 0





The Toledo invariant in this case is 2(g − 1). All the ﬂat U(2, 1)-connections

with τ = 2(g − 1) are reducible by Proposition 5.1. The fact that there

is no irreducible deformation for the U(2, 1)-connections with τ = 2(g − 1)

was ﬁrst demonstrated by Toledo [26]. In particular, this component is

connected [6, 11].

7. Hodge Bundles and Deformation.

Let p = 2 and q = 1 and assume τ ≥ 0 and 0 ≤ c < 3. A Hodge bundle on

X is a direct sum of holomorphic bundles [22]

V =

s,t

together with holomorphic maps (Higgs ﬁeld)

: V

s,t

−→ V

s−1,t+1

⊗ Ω.

An immediate consequence of Proposition 3.1 is:

Corollary 7.1. Suppose (V

⊕ V

, (Φ

, Φ

)) ∈ M

Dol

of Proposition 3.1). Then (V

⊕ V

, (Φ

, Φ

)) is a Hodge bundle if and only

if (V

⊕ V

, (Φ

, Φ

)) is either binary or t ernary in the following sense:

1) Binary: Φ

≡ 0.

2) Ternary: V

= V

⊕ V

and the Higgs ﬁeld consists of two maps:

: V

−→ V

⊗ Ω,

: V

−→ V

⊗ Ω.

Denote by B(d

, d

) the space of all poly-stable (or S-equivalence classes

of semi-stable) binary Hodge bundles (V

⊕ V

, (Φ

, 0)) with deg(V

) = d

and deg(V

) = d

. Denote by T (d

, d

) the space of all poly-stable

(or S-equivalence classes of semi-stable) ternary Hodge bundles (V

⊕ V

⊕

, (Φ

, Φ

)) with deg(V

) = d

, deg(V

) = d

and deg(V

) = d

. Denote

the subspaces of stable Hodge bundles by B(d

, d

)

, T (d

, d

)

. When

τ is not an integer, these are the type (2,1) and (1,1,1) spaces in [8]. Note

the (1,2) typ e s give τ < 0 and therefore need not be considered here.

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 245

Proposition 7.2. Every stable binary Hodge bundle in (M

Dol

)

may be

deformed to a stable ternary Hodge bundle within M

Dol

A family (or ﬂat family) of Higgs pairs (V

, Φ

) is a variety Y such

that there is a vector bundle V

on X × Y together with a section Φ

∈

Γ(Y, (π

)

∗

(π

∗

Ω⊗End(V

))) [18]. CM

Dol

being a moduli space implies that

if Y is a family of stable (poly-stable or S-equivalence classes of semi-stable)

Higgs bundles, then there is a natural morphism [15, 17]

t : Y −→ CM

Dol

Moreover t takes every point y ∈ Y to the point of CM

Dol

that corresponds

to the Higgs bundle in the family over y [15, 17, 18].

The s pace M

Dol

(c); hence, to show that two

stable (poly-stable or S-equivalence classes of semi-stable) Higgs bundles

, Φ

) and (V

, Φ

) belong to the same component of M

Dol

(c), it suﬃces

to exhibit a connected family Y (within M

Dol

(c)) of stable (poly-stable or

S-equivalence classes of semi-stable) Higgs bundles containing both (V

, Φ

)

and (V

, Φ

Proof. Suppose (V, Φ) = (V

⊕ V

, (Φ

, 0)) ∈ B(d

, d

)

⊂ (M

Dol

)

. Since

s(V

) ≥ s(V ) (This is due to the assumption τ ≥ 0, and 0 ≤ c < 3), Φ

6≡ 0.

Construct the canonical factorization for Φ

0 −→ V

−→ V

−→ 0



0 ←− W

←− V

⊗ Ω

←− W

←− 0 .

being Φ

invariant implies

deg(V

) = s(V

) < s(V ) ≤ s(V

) ≤ s(V

) = deg(V

The space P ic

(X) of line bundles of degree 0 over X is identiﬁed with

the Jacobi variety J

(X). Construct the universal bundle [2, 19]

U −→ X × J

(X)

such that U restricts to the bundle L ⊗ V

⊗ V

−1

on (X, L). Let π be the

projection

π : X × J

(X) −→ J

(X).

Applying the right derived functor R

to π gives the sheaf F = R

∗

(U)

[10] such that

= H

(X, L ⊗ V

⊗ V

−1

Since

deg(L ⊗ V

⊗ V

−1

) = deg(V

) − deg(V

) < 0,

246 EUGENE Z. XIA

by Riemann-Roch,

(L ⊗ V

⊗ V

−1

) = h

(L ⊗ V

⊗ V

−1

) − deg(L ⊗ V

⊗ V

−1

) + (g − 1)

= deg(V

) − deg(V

) + (g − 1)

is a constant. By Grauert’s theorem, F is locally free, hence, is associated

with a vector bundle

F 7−→ J

(X)

of rank deg(V

)−deg(V

)+(g−1). In particular the total space F is smooth

and parameterizes extensions [9, 10]:

0 −→ L ⊗ V

−→ W

−→ V

−→ 0

for ﬁxed V

, V

. Tensoring the above sequence with Ω gives:

0 −→ L ⊗ V

⊗ Ω

−→ W

⊗ Ω

−→ V

⊗ Ω −→ 0.

Fix ϕ. Then F also parameterizes a family of Higgs bundles (W

, Φ

) that

ﬁt into the factorization

0 −→ L ⊗ V

−→ W

−→ V

−→ 0



0 ←− W

←− V

⊗ Ω

←− W

←− 0 .

Let V ⊂ F be the subset of stable extensions (i.e., W

∈ V implies W

a stable holomorphic bundle [19]).

Lemma 7.3. V ∩ H

(L ⊗ V

⊗ V

−1

) and V are open and dense in H

(L ⊗

⊗V

−1

) and F , respectively. Moreover if W

∈ V, then (W

⊕V

, (Φ

, 0))

is stable.

Proof. Since deg(L ⊗ V

) < deg(V

) for each L ∈ J

(X), by a theorem of

Lange and Narasimhan [13], there always exists a stable extension W

∈

(L ⊗ V

⊗ V

−1

). In addition, a theorem of Maruyama states that being

stable is an open prope rty [14]. The open dense property follows from the

smoothness of F and H

(L ⊗ V

⊗ V

−1

Let p

, p

be the projections of W

⊕ V

onto its W

and V

factors,

respectively. Suppose W is (Φ

, 0)-invariant. Suppose W has rank 1. If

(W ) = 0, then W = L ⊗ V

; otherwise, deg(W ) ≤ deg(V

). In either

case, s(W ) < s(V ). Suppose W has rank 2. If p

(W ) = 0, then W = W

and s(W ) < s(V ). Suppose P

(W ) 6= 0. Then there exists a line bundle L

such that

0 −→ L

−→ W

−→ p

(W ) −→ 0.

Now let L

= p

) ⊂ W

. Then

deg(W ) = deg(L

) + deg(p

(W )) ≤ deg(L

) + deg(V

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 247

Since W

is stable, s(L

) < s(W

). By the assumptions τ ≥ 0 and 0 ≤ c <

3, one has s(V

) ≤ 0 and s(W

) ≥ 0. Therefore,

s(W ) ≤ s(L

⊕ V

) =

s(L

) + s(V

)

s(W

) + s(V

)

deg(W

)

deg(V

)

≤

deg(W

) + deg(V

)

= s(V ).

Thus (W

⊕ V

, (Φ

, 0)) is stable. 

Since Φ

6≡ 0, deg(V

) ≤ d

+ 2(g − 1) and

deg(V

) = d

− deg(V

) ≥ d

− d

− 2(g − 1).

Hence

deg(V

−1

⊗ V

⊗ Ω) = −d

+ deg(V

) + 2(g − 1) ≥ d

− 2d

> 0.

Hence there exists L

∈ J(X) such that

−1

⊗ L

⊗ V

⊗ Ω) > 0

implying there exists a non-trivial holomorphic map

φ : V

−→ L

⊗ V

⊗ Ω.

Fix φ 6≡ 0. By Lemma 7.3, the family parameterized by V contains both

⊕ V

, (Φ

, 0)) and (W

⊕ V

, (Φ

, 0)) implying there is deformation

between the two.

Set L = L

and Φ

= g

◦ φ. Then the family of stable Higgs bundles

parameterized by H

(X, V

−1

⊗ L

⊗ V

⊗ Ω) contains (W

⊕ V

, (Φ

, 0)) and

⊕ V

, (Φ

, Φ

)).

Now the family of bundle extensions of V

by L

⊗V

is H

⊗V

−1

With a ﬁxed φ and the canonical factorization with ϕ ﬁxed, H

⊗ V

⊗

−1

) parameterizes a family of Higgs bundles. This family contains (W

⊕

, (Φ

, Φ

)). The zero element in H

⊗ V

−1

) corresponds to the

bundle extension

0 −→ L

⊗ V

−→ (L

⊗ V

) ⊕ V

−→ V

−→ 0.

Tensoring with Ω gives

0 −→ L

⊗ V

⊗ Ω

−→ ((L

⊗ V

) ⊕ V

) ⊗ Ω

−→ V

⊗ Ω −→ 0.

Lemma 7.4. If (W

⊕ V

, (Φ

, Φ

)) is stable (semi-stable), then H

⊗

⊗ V

−1

) parameterizes a stable (semi-stable) family.

Proof. Suppose (U

⊕ V

, (Ψ

, Ψ

)) ∈ H

⊗ V

−1

) and W ⊂ U

⊕ V

is (Ψ

, Ψ

)-invariant. Since ϕ, φ 6≡ 0, one has W = V

or W = V

⊕ V

. A

direct computation shows s(W ) < s(U

⊕ V

) (s(W ) ≤ s(U

⊕ V

)). 

248 EUGENE Z. XIA

Proposition 7.2 follows from Lemma 7.4 because the family of Higgs bun-

dles parameterized by H

⊗ V

−1

) contains (W

⊕ V

, (Φ

, Φ

)) and

((L

⊗ V

) ⊕ V

⊕ V

, (g

◦ ϕ ◦ f

, g

◦ φ)).

To summarize, a stable binary Hodge bundle (V

⊕ V

, (Φ

, 0)) is ﬁrst

deformed to (W

⊕V

, (Φ

, 0)) such that non-trivial holomorphic maps exist

between V

and (L

⊗ V

) ⊗ Ω ⊂ W

⊗ Ω. Such a non-trivial map Φ

is then

chosen and attached to the existing Higgs ﬁeld Φ

. Finally W

is deformed

to a direct sum making the resulting stable Higgs bundle a ternary Hodge

bundle. 

Let B = B(0, 0)\(B(0, 0)

∪ T (0, 0, 0)).

Proposition 7.5. B is connected and can be deformed to a stable ternary

Hodge bundle in M

Dol

Proof. Consider the space U × J

(X), where J

(X) is the Jacobi variety

identiﬁed with the s et of holomorphic line bundles of degree zero on X and

U is the moduli space of rank-2 poly-stable holomorphic bundles of degree 0

on X. The space U is connected [2, 19]. Hence U ×J

(X) is connected. Each

poly-stable Higgs bundle in B is contained in the family of Higgs bundles

parameterized by U × J

(X). Hence the natural morphism

t : U × J

(X) −→ B

is surjective. This proves that the set B is connected.

Choose holomorphic line bundles V

, V

of degrees −1, 1, 0, respec-

tively such that

(X, V

−1

⊗ V

⊗ Ω) > 0,

(X, V

−1

⊗ V

⊗ Ω) > 0.

Choose

0 6≡ ψ

∈ H

(X, V

−1

⊗ V

⊗ Ω)

0 6≡ ψ

∈ H

(X, V

−1

⊗ V

⊗ Ω).

The space of extension of V

by V

0 −→ V

−→ V

−→ 0,

is H

(X, V

⊗ V

−1

). Tensoring the exac t sequence with Ω gives

0 −→ V

⊗ Ω

−→ V

⊗ Ω

−→ V

⊗ Ω −→ 0.

Since deg(V

) < deg(V

), by the theorem of Lange and Narasimhan [13],

stable extensions always exist. Fix a stable extension V

and set

= ψ

◦ f

= g

◦ ψ

Note (V

⊕ V

, 0) ∈ B. The connected family

F C = H

(X, V

−1

⊗ V

⊗ Ω) × H

(X, V

−1

⊗ V

⊗ Ω)

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 249

of Higgs bundles contains (V

⊕ V

, 0) and (V

⊕ V

, (Φ

, Φ

)). Note the

family F C contains s emi- stable Higgs bundles. This is allowed since the

points in the moduli space M

Dol

are also interpreted as S-equivalence classes

of sem i-stable Higgs bundles. However one may choose F C to be a strictly

poly-stable family:

F C = (H

(X, V

−1

⊗ V

⊗ Ω) × H

(X, V

−1

⊗ V

⊗ Ω)) \

(({0} × H

(X, V

−1

⊗ V

⊗ Ω)) ∪ (H

(X, V

−1

⊗ V

⊗ Ω) × {0})).

Since V

is stable, by Lemma 7.3, any element in F C is semi-stable. Hence

the family F C provides a deformation between (V

⊕ V

, 0) and (V

⊕

, (Φ

, Φ

)). The cohomology H

(X, V

⊗ V

−1

) parameterizes bundle ex-

tensions of V

by V

and also parameterizes a family of Higgs bundles

with ﬁxed ψ

, ψ

. By Lemma 7.4, this is a stable family which contains

⊕ V

, (Φ

, Φ

)) and (V

⊕ V

, (ψ

◦ f

, g

◦ ψ

)) where f

, f

, g

come from the trivial extensions

0 −→ V

−→ V

⊕ V

−→ V

−→ 0,

0 −→ V

⊗ Ω

−→ (V

⊕ V

) ⊗ Ω

−→ V

⊗ Ω −→ 0.

Hence H

(X, V

⊗ V

−1

) provides a deformation between (V

⊕ V

, (Φ

, Φ

))

and (V

⊕ V

, (ψ

◦ f

, g

◦ ψ

)) ∈ T (−1, 1, 0).

To summarize, one ﬁrst shows that the space B is connected. Then choose

a speciﬁc element (V

⊕ V

, 0) ∈ B with V

a stable extension of V

by V

and that there exists non-trivial holomorphic maps

: V

−→ V

⊗ Ω

: V

−→ V

⊗ Ω.

This provides a deformation from (V

⊕V

, 0) to (V

⊕V

, (Φ

, Φ

)). Finally,

since V

is an extension of V

by V

, (V

⊕ V

, (Φ

, Φ

)) is deformed to

⊕ V

, (ψ

◦ f

, g

◦ ψ

)) in H

(X, V

⊗ V

−1

). 

Corollary 7.6. Every Binary Hodge bundle can be deformed to a ternary

Hodge bundle.

Proof. Every poly-stable reducible Hodge bundle is either ternary or in B.

The result then follows from Proposition 7.2 and 7.5. 

Lemma 7.7. For ﬁxed integers d

, d

, T (d

, d

) is connected.

Proof. We ﬁrst consider the stable bundles. Stability implies the Higgs ﬁelds

, Φ

are not identically zero. Denote by J

(X) the Jacobi variety identiﬁed

with the set of holomorphic line bundles of degree d. For each L

∈ J

(X),

the set of all (L

, Φ

) such that L

∈ J

(X) and

0 6≡ Φ

∈ H

(X, L

−1

⊗ L

⊗ Ω)

250 EUGENE Z. XIA

is C

∗

× Sym

+2(g−1)−d

X, where Sym

X is the d-th symmetric product of

X. Hence the set of all triples (L

, L

, Φ

) such that

7−→ L

⊗ Ω

with Φ

6≡ 0 is the space (C

∗

× Sym

+2(g−1)−d

X) × J

(X).

Similarly, for each L

∈ J

(X), the space of all triples (L

, L

, Φ

) such

that

7−→ L

⊗ Ω

with Φ

6≡ 0 is C

∗

× Sym

+2(g−1)−d

X. The set of Higgs bundles parame-

terized by the total space

S = (C

∗

× Sym

+2(g−1)−d

X) × (C

∗

× Sym

+2(g−1)−d

X) × J

(X)

contains every Higgs bundle in T (d

, d

). Hence the natural morphism

t : S −→ T (d

, d

)

is surjective. Since S is connected, T (d

, d

) is connected.

The reducible bundles consist of T (0, 0, 0) and T (0, d

, −d

). All poly-

stable Higgs bundles associated with the points in T (0, 0, 0) and T (0, d

, −d

)

are contained in the families parameterized by

= J

(X) × J

(X)

and

= (C

∗

× Sym

2(g−1)−2d

X) × J

−d

(X) × J

(X),

respectively. Both S

, S

are connected. Since the natural morphisms

: S

−→ T (0, 0, 0)

: S

−→ T (0, d

, −d

)

are surjective, both T (0, 0, 0) and T (0, d

, −d

) are connected. 

Proposition 7.8. Every component of M

Dol

contains a Hodge bundle.

Proof. By Corollary 4.8, every component of M

Dol

contains a local minimum

(V, Φ) of m

. If (V, Φ) is a smooth point, then (V, Φ) is a critical point of m.

A theorem of Hitchin and Simpson implies that (V, Φ) is a Hodge bundle

[12, 22]. Singular points of M

Dol

correspond to reducible Higgs bundles.

The space of all reducible Higgs bundles correspond to either the space of

U(2) × U(1) representations or the space of U(1) × U(1, 1) representations.

Each component of U(2) × U(1) and U(1) × U(1, 1) representations contains

points that correspond to Hodge bundles [11]. In fact, these points are

exactly the ones corresponding to the points in B and T (0, d

, −d

). 

Let K be a divisor of Ω and let

w : X −→ |K|

∼

g−1

be the canonical map [10].

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 251

Lemma 7.9. Ω has a section with simple zeros.

Proof. The linear system |K| is base point free [10]. If X is hyperelliptic,

then the map w is a 2-1 branch map into CP

g−1

and an embedding otherwise .

In both cases, by Bertini’s theorem, there exists a hyperplane H ∈ CP

g−1

such that H ∩ X is regular. Then w

−1

(H) is an eﬀective divisor equivalent

to K and with simple zeros. 

Choose

K = {x

, x

, . . . , x

2(g−1)

such that the x

’s are all distinct.

Proposition 7.10. Let 0 ≤ τ < 2(g − 1). Suppose

T (d

− 1, d

+ 1, d

), T (d

, d

) ⊂ M

Dol

Then there is deformation between T (d

, d

) and T (d

− 1, d

+ 1, d

)

within M

Dol

Proof. Suppose

⊕ V

, (Φ

, Φ

)) ∈ T (d

− 1, d

+ 1, d

⊕ U

, (Ψ

, Ψ

)) ∈ T (d

, d

By the semi-stability of (U

⊕ U

, (Ψ

, Ψ

)) and the assumptions τ ≥

0, 0 ≤ c < 3, one has d

≤ 0 and

− 1 < d

≤

+ d

< 1;

hence,

− 1 < d

≤ 0 and d

+ 1 > 0.

This implies (V

⊕ V

, (Φ

, Φ

)) is stable. Hence Φ

6≡ 0 and

− deg(V

) + d

+ 2(g − 1) ≥ 0.

On the other hand, deg(V

) + deg(V

) = d

, so

− deg(V

) − d

≤ 2(g − 1),

−d

< 1 − d

= − deg(V

) ≤ −d

+ d

+ 2(g − 1) ≤ 2(g − 1).

In light of Lemma 7.7, it suﬃces to demonstrate the existence of (U

⊕

⊕ U

, (Ψ

, Ψ

)) ∈ T (d

, d

) and (V

⊕ V

⊕ U

, (Φ

, Φ

)) ∈ T (d

−

1, d

+ 1, d

) and a deformation between the two.

Since |K| is base point free, there exists K

∈ |K| such that

= {y

, y

, . . . , y

2(g−1)

}

252 EUGENE Z. XIA

with y

6= x

2(g−1)

for all 1 ≤ i ≤ 2g. The b ounds on the degrees of the

various bundles allow us to construct the following divisors:











= {−x

, . . . , −x

− deg(U

)

}

= {y

, . . . , y

−deg(V

)

, −x

2(g−1)

}

= {−y

−deg(V

)+1

, . . . , −y

−deg(V

)−d

Let u be the basic epimorphism [1]

u : Div (X) −→ H

(X, O

∗

)

and set











= u(D

)

= u(D

)

= u(D

)

= U

⊕ U

Let ψ

, ψ

be meromorphic sections associated with the divisors D

, D

Then the meromorphic section ψ

⊕ ψ

of U

is associated with the divisor

= {−x

, . . . , −x

− deg(U

)

, −x

2(g−1)

Hence there exists V

⊂ U

[9] such that

= u(D

Let

= U

Since

⊗ V

= det(U

) = U

⊗ U

= u(D

where

= {y

, . . . , y

−deg(V

)

In short, the bundle U

is constructed in such a way that it is the trivial

extension of U

by U

, and is also an extension of V

by V

0 −→ U

−→ U

−→ 0

0 −→ V

−→ U

−→ V

−→ 0.

Tensoring with Ω gives

0 −→ U

⊗ Ω

−→ U

⊗ Ω

−→ U

⊗ Ω −→ 0

0 −→ V

⊗ Ω

−→ U

⊗ Ω

−→ V

⊗ Ω −→ 0.

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 253

Since











−D

+ D

+ K

2(g−1)

, y

−deg(V

)−d

, . . . , y

2(g−1)

−D

+ D

+ K =

−deg(V

)+1

, . . . , y

−deg(V

)−d

− deg(U

)+1

, . . . , x

2(g−1)

are eﬀective divisors, there exists

0 6≡ ψ

∈ H

(X, U

−1

⊗ U

⊗ Ω)

0 6≡ ψ

∈ H

(X, U

−1

⊗ U

⊗ Ω).

Set

= ψ

◦ f

and Ψ

= g

◦ ψ

Then (U

⊕ U

, (Ψ

, Ψ

)) is a semi-stable ternary Hodge bundle.

The divisors











−D

+ D

+ K

−deg(V

)−d

, . . . , y

2(g−1)

−D

+ D

+ K =

− deg(U

)+1

, . . . , x

2(g−1)−1

−deg(V

)+1

, . . . , y

−deg(V

)−d

are eﬀective. Hence there exist

0 6≡ φ

∈ H

(X, V

−1

⊗ U

⊗ Ω)

0 6≡ φ

∈ H

(X, U

−1

⊗ V

⊗ Ω).

Remark 2. This is the critical step where the assumption τ < 2(g − 1) is

needed. In the case of τ = 2(g − 1), the degree of V

−1

⊗ U

⊗ Ω equals

−1 thus rendering it impossible to ﬁnd a non-zero global section φ

. This

reﬂects the fact that every representation with τ = 2(g − 1) is reducible.

(See Section 6.)

Set

= φ

◦ f

and Ψ

= g

◦ φ

Then (U

⊕ U

, (Ψ

, Ψ

)) is a semi-stable Higgs bundle. Since

(X, U

−1

⊗ U

⊗ Ω) > 0

(X, U

−1

⊗ U

⊗ Ω) > 0,

(X, U

−1

⊗ U

⊗ Ω) and H

(X, U

−1

⊗ U

⊗ Ω) are proper subspaces of

(X, U

−1

⊗ U

⊗ Ω) and H

(X, U

−1

⊗ U

⊗ Ω), respectively. Hence

F C = (H

(X, U

−1

⊗ U

⊗ Ω) \ H

(X, U

−1

⊗ U

⊗ Ω)) ×

(X, U

−1

⊗ U

⊗ Ω) \ H

(X, U

−1

⊗ U

⊗ Ω))

254 EUGENE Z. XIA

is connected and parameterizes a family of sem i-stable Higgs bundles that

contains both (U

⊕ U

, (Ψ

, Ψ

)) and (U

⊕ U

, (Ψ

, Ψ

)). Hence there is

deformation between the two.

The space of bundle extensions of V

by V

0 −→ V

−→ V

−→ 0,

is parameterized by the vector space H

⊗ V

−1

) containing both U

and

⊕ V

(the zero element in H

⊗ V

−1

)). Again tensoring with Ω gives

0 −→ V

⊗ Ω

−→ V ⊗ Ω

−→ V

⊗ Ω −→ 0.

Let

= φ

◦ f

and Φ

= g

◦ φ

where

0 −→ V

−→ V

⊕ V

−→ V

−→ 0

0 −→ V

⊗ Ω

−→ (V

⊕ V

) ⊗ Ω

−→ V

⊗ Ω −→ 0

correspond to the trivial extensions. By Lemma 7.4, H

⊗ V

−1

) param-

eterizes a family of semi-stable Higgs bundles that contains both (U

⊕

, (Ψ

, Ψ

)) and (V

⊕ V

⊕ U

, (Φ

, Φ

)).

To summarize, the ﬁrst step consists of ﬁxing U

= U

⊕ U

and deform

the Higgs ﬁeld (Ψ

, Ψ

) to (Ψ

, Ψ

). In the second step, ﬁx φ

, φ

and deform

to V

⊕ V



Consider the space T (0, d

, −d

). By Proposition 7.5, one may assume d

0. To deform points in T (0, d

, −d

), the family F C constructed in the

above proof contains semi-stable Higgs bundles. However, one may also opt

to construct the deformation family of poly-stable Higgs bundles by setting:

F C = (H

(X, U

−1

⊗ U

⊗ Ω) \

(X, U

−1

⊗ U

⊗ Ω) ∪ H

(X, U

−1

⊗ U

⊗ Ω))) ×

(X, U

−1

⊗ U

⊗ Ω) \

(X, U

−1

⊗ U

⊗ Ω) ∪ H

(X, U

−1

⊗ U

⊗ Ω)))

∪(H

(X, U

−1

⊗ U

⊗ Ω) × {0}).

The case with τ = 2(g − 1) has been covered in Section 6 and M

2(g−1)

Dol

is connected. Suppose τ < 2(g − 1). By Proposition 7.8, every component

of M

Dol

contains a Hodge bundle. By Corollary 7.6, every component of

Dol

contains a ternary Ho dge bundle. It follows from Proposition 7.10

and induction that M

Dol

is connected. Since

Dol

= M

Dol

(X, O

∗

Theorem 1.1 then follows from Corollary 4.2.

FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES 255

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Received September 2, 1998 and revised March 16, 1999.

University of M assachusetts

Amherst, MA 010 03 -45 15

E-mail address: xia@math.umass.edu

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PACIFIC JOURNAL OF MATHEMATICS

Volume 195 No. 1 September 2000

Motion of hypersurfaces by Gauss curvature 1

BEN ANDREWS

Summation of formal solutions of a class of linear difference equations 35

B.L.J. BRAAKSMA, B.F. FABER AND G.K. IMMINK

Approximation of recurrence in negatively curved metric spaces 67

CHARALAMBOS CHARITOS AND GEORGIOS TSAPOGAS

Cohomology of singular hypersurfaces 81

BERNARD M. DWORK

Riemannian manifolds admitting isometric immersions by their ﬁrst

eigenfunctions 91

AHMAD EL SOUFI AND SAÏD ILIAS

The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set

of a stable map 101

TSUYOSHI KOBAYASHI AND OSAMU SAEKI

Geometric realizations of Fordy–Kulish nonlinear Schr

odinger systems 157

JOEL LANGER AND RON PERLINE

Seiberg–Witten invariants for 3-manifolds in the case b

= 0 or 1 179

YUHAN LIM

Virtual homology of surgered torus bundles 205

JOSEPH D. MASTERS

Group actions on polynomial and power series rings 225

PETER SYMONDS

The moduli of ﬂat PU(2,1) structures on Riemann surfaces 231

EUGENE Z. XIA

0030-8730(200009)195:1;1-#

Paciﬁc Journal of Mathematics 2000 Vol. 195, No. 1

Paciﬁc

Journal of

Mathematics

Volume 195 No. 1 September 2000

Anisotropic flow, entropy, and -Minkowski problem

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We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak’s $L^p$ -Minkowski problem on $S^n$ which were obtained by other methods.

Anisotropic flow, entropy and $L^p$-Minkowski problem

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We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak's $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

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We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $\mu$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$.

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We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in R n + 1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

L^p$-Minkowski Problem under Curvature Pinching

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Jul 2023

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$ is comparable to the standard Euclidean metric of $\mathbb{R}^{n}$ up-to a factor of $\gamma > 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_\gamma := 1 - \frac{n+1}{\gamma}$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma < \infty$. In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.

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We consider translating solitons to flows by positive powers α of the Gaussian curvature, called Kα-flows, in Riemannian products M×R. We prove that, when M is the Euclidean space Rn, the sphere Sn, or one of the hyperbolic spaces HFm, there exist complete rotational translating solitons to Kα-flow in M×R for certain values of α.

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L p -Minkowski Problem Under Curvature Pinching

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Entropy estimates for evolving hypersurfaces

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Ben Andrews

We consider parabolic curvature flows for hypersurfaces in Euclidean space, defined in terms of the mixed discriminants of Aleksandrov. Dilation-invariant curvature integral estimates are proved for convex hypersurfaces evolving by these equations, generalising the so-called entropy estimates which have been proved for special cases by Richard Hamilton, Ben Chow, and the author. The earlier proof by contradiction is replaced here by a new proof which is more direct and applies more generally, and which demonstrates the intimate connection between entropy estimates and the Aleksandrov-Fenchel inequalities.

The heat equation shrinks embedded plane curves to round points

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J DIFFER GEOM

Matthew A. Grayson

Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est lisse pour tout t, il converge vers un point quand t\T et sa forme limite quand t→T est un cercle rond, avec convergence dans norme C ∞

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Gerhard Huisken

Elementary acceleration and multisummability. I

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Minimal Varieties in Riemannian Manifolds

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Richard S. Hamilton

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Weimin Chen

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