Null hypersurfaces evolved by their mean curvature in a Lorentzian manifold

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COLLOQUIUM MATHEMATICUM
Online First version
NULL HYPERSURFACES EVOLVED BY THEIR
MEAN CURVATURE IN A LORENTZIAN MANIFOLD
BY
FORTUNÉ MASSAMBA and SAMUEL SSEKAJJA (Scottsville)
Abstract. We use null isometric immersions to introduce time-dependent null hy-
persurfaces, in a Lorentzian manifold, evolving in the direction of their mean curvature
vector (a vector transversal to the null hypersurface). We prove an existence result for
such hypersurfaces in a short-time interval. Then, we discuss the evolution of some in-
duced geometric objects. Consequently, we prove under certain geometric conditions that
some of the above objects will blow-up in finite time. Also, several examples are given to
illustrate the main ideas.
1. Introduction. Flow of Riemannian hypersurfaces by functions of
their mean curvatures (for example, mean curvature flow, inverse mean cur-
vature flow, and many more) has been an interesting area of research in the
past 30 years. Numerous results have been obtained: for instance, Huisken
[H98] showed that inverse mean curvature evolution can be used to relate
the size of a black hole and its total energy. On the other hand, the same au-
thor [H84] showed that the solution to mean curvature evolution with some
specified initial data remains smooth, compact and convex until it shrinks
to a “round point” in a finite time; that is, the asymptotic shape of the
evolving hypersurface just before it disappears is a sphere. Gage and Hamil-
ton [GH86] used curve shortening flow to prove on isoperimetric inequality
for convex planar domains, and Andrews [A97] used affine mean curvature
flow to prove affine isoperimetric inequalities. More results and details on
Riemannian mean curvature evolution can be found in the above mentioned
papers and references therein.
Lightlike hypersurfaces are closely linked to black hole theory and the
study of trapped surfaces. In fact, it has been shown in [DB96] that black hole
horizons can be represented by null hypersurfaces. The classical approach
to the above phenomena has been the use of expansion scalars (null mean
curvature) from null geodesics, via Raychaudhuri’s equation (see [DS10] for
2010 Mathematics Subject Classification: Primary 53C50; Secondary 53C44, 53C40.
Key words and phrases: null hypersurface, null mean curvature flow, maximum principle.
Received 7 November 2017.
Published online 1 March 2019.
DOI: 10.4064/cm7450-8-2018 [1] c
Instytut Matematyczny PAN, 2019

2F. MASSAMBA AND S. SSEKAJJA
details). Literature shows that our universe is expanding in time. This can
be supported by the fact that very distant stars appear with red-shifts. By
definition, a red-shift is a change in color when an object that emits light is
moving away from the observer [DS10]. It is called a red-shift because the
shifts are towards the red part of the spectrum. Heavenly bodies like stars
are moving away from us radially, so we say that at one time the universe was
smaller. Going back far enough in time, it is evident from the above explana-
tion that the universe may have been something the size of an atom, what we
call the Primordial Atom [DS10]. Along with the expanding universe, since
the black holes are surrounded by a local mass distribution and expand by
the inflow of galactic debris as well as electromagnetic and gravitational ra-
diation, their area increases in a given physical situation. Hence black holes
are also expanding in time. Therefore, although the classical isolated black
holes have been extensively studied (see [DS10] and references therein), they
do not represent a realistic model in the context of an expanding universe. To
address this issue of representing expanding black holes, Ashtekar–Krishnan
[AK03] introduced a concept of dynamical horizons which are a special type
of spacelike hypersurfaces of a spacetime whose asymptotic states are the
isolated horizons.
Motivated by the fact that our universe (and its constituents) is changing
with time, we introduce a geometric evolution of null hypersurfaces along
their null mean curvature. In particular, this kind of evolution has a potential
to explain, among other applications, the idea of trapped surfaces (hence
the study of black holes since trapped surfaces are linked to black holes).
To the best of our understanding, this is the first piece of work in this
direction on null hypersurfaces in Lorentzian manifolds, despite the fact that
the study of null hypersurfaces was started long time ago by Duggal–Bejancu
[DB96] and Kupeli [K96], and up-to-date results may be found in the books
[DB96, DS10]. This motivated other researchers to investigate the geometry
of null submanifolds (see for example [A09, AET03, DL14, J13, MS16]).
The paper is arranged as follows: In Section 2 we quote some basic geo-
metric tools necessary for the development of other sections. In Section 3 we
define the concept of null mean curvature flow (MCF) for null hypersurfaces,
using null isometric immersions (see Definition 3.2), and give some support-
ing physical example. Section 4 is devoted to existence results and geometry
of induced objects on the evolving hypersurface.
2. Preliminaries. Consider an (n+ 1)-dimensional null hypersurface
(M, g),n≥2, of an (n+ 2)-dimensional Lorentzian manifold (M, g), where
gis a degenerate metric on M, induced by the Lorentzian metric gof M.
In the null hypersurface case, basic differences occur mainly due to the fact
that the normal vector bundle T M ⊥is the same as the null tangent bundle

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 3
along a non-zero differentiable radical distribution Rad T M of M, defined
by
Rad TpM=TpM⊥={Ep∈TpM:g(Ep, X)=0,∀X∈TpM},
where dim(Rad T M ) = 1. There exists a Riemannian screen distribution
[DB96], denoted S(T M ), on Mwhich is complementary to the radical distri-
bution such that we have the orthogonal direct sum T M =T M ⊥⊥S(T M ).
Throughout this paper, Γ(Ξ)will denote the F(M)-module of differentiable
sections of a vector bundle Ξ. The manifolds we consider are supposed to
be paracompact, smooth and connected. Therefore, the existence of S(T M )
is secured. However, in general, S(T M )is not canonical (thus not unique)
and the null geometry depends on its choice. But it is known [DB96, DS10]
that S(T M )is canonically isomorphic to the bundle T M/T M ⊥considered
by Kupeli [K96].
From [DB96, p. 79, Theorem 1.1], we know that for a screen distribution
S(T M )on Mthere exists a unique vector bundle tr(T M )such that for any
non-zero local normal section E∈Γ(Rad T M |U)on U ⊂ Mthere exists a
unique section Nof tr(T M )|Usatisfying g(E, N ) = 1 and g(N , Z)=0for all
Z∈Γ(S(T M )|U). Then we have the decomposition T M |M=T M ⊕tr(T M ).
Let ∇and ∇∗denote the induced connections on Mand S(T M )re-
spectively, and Pbe the projection of T M onto S(T M ). Then the local
Gauss–Weingarten equations of Mand S(T M )are the following [DB96]:
∇XY=∇XY+h(X, Y ) = ∇XY+B(X, Y )N,(2.1)
∇XN=−ANX+∇t
XN=−ANX+τ(X)N,(2.2)
∇XP Y =∇∗
XP Y +h∗(X , P Y ) = ∇∗
XP Y +C(X , P Y )E,(2.3)
∇XE=−A∗
EX+∇∗t
XE=−A∗
EX−τ(X)E, A∗
EE= 0,(2.4)
for all X, Y ∈Γ(T M ),E∈Γ(T M ⊥)and N∈Γ(tr(T M )), where ∇is the
Levi-Civita connection on M. In the above setting, Bis the local second fun-
damental form of M, and Cis the local second fundamental form on S(T M ).
ANand A∗
Eare the shape operators on T M and S(T M )respectively, while
τis a 1-form on T M . The above shape operators are related to their local
fundamental forms by
g(A∗
EX, Y ) = B(X, Y ), g(ANX, P Y ) = C(X , P Y ),(2.5)
g(A∗
EX, N )=0, g(ANX, N )=0,∀X, Y ∈Γ(T M ).(2.6)
From (2.6) we notice that A∗
Eand ANare both screen-valued operators.
Let ϑ=g(N, ·)be a 1-form metrically equivalent to Ndefined on M.
Take λ=i∗ϑto be its restriction on M, where i:M→Mis the inclusion
map. Then it is easy to show that
(2.7) (∇Xg)(Y, Z) = B(X, Y )λ(Z) + B(X,Z)λ(Y),∀X, Y, Z ∈Γ(T M ),

4F. MASSAMBA AND S. SSEKAJJA
which indicates that ∇is generally not a metric connection with respect
to g. However, the induced connection ∇∗on S(T M )is a metric connection.
For more details about null hypersurfaces see the books [DB96] and [DS10].
The degeneracy of the induced metric tensor gon Mis associated with
several challenges in the study of null geometry (for instance see [DB96] and
[DS10]). Prior to these books, Katsuno [K81] introduced a non-degenerate
metric (called the associated metric) on null hypersurfaces in 4-dimensional
Lorentzian manifolds. This metric was extended to null hypersurfaces in
(n+ 2)-dimensional Lorentzian manifolds in [A09] and used to define some
geometric operators on null hypersurfaces. It has also been linked to null
rigging techniques on null hypersurfaces (for example, see [GO16] and other
references therein).
More precisely, for a null hypersurface (M, g)of a Lorentzian manifold
(M , g), the associated metric bgis given by
bg(X, Y ) = g(X, Y ) + λ(X)λ(Y),∀X, Y ∈Γ(T M ).(2.8)
The metric bgis invertible and its inverse, g[·,·], was called the pseudo-inverse
of g(see [AET03]). Also, observe that bgcoincides with gif the latter is
non-degenerate. The metric bghas been used to define (on M) the usual
operators such as gradient,divergence,d’Alembertian (see [AET03]), which
one cannot afford with the degenerate metric g. In case bgcoincides with g
on M, we define the gradient ∇sf, Hessian Hesss(f), and d’Alembertian ∆sf
of a smooth function fon U ⊂ Mwith respect to the screen distribution
S(T M )as
(2.9) ∇sf=gαβXα(f)Xβ,Hesss(f) = Xα(Xβ(f)) −(∇∗
XαXβ)(f),
∆s(f) = trs(Hesss(f)) = gαβ(Xα(Xβ(f)) −(∇∗
XαXβ)(f)),
where {X1, . . . , Xn}is a basis of S(T M ), and trs(·)denotes the trace with
respect to S(T M ).Throughout this paper, we assume that Mcarries the
associated metric bg, and tr(·)will denote the trace over Mwith respect to bg.
Denote by Rand Rthe curvature tensors of Mand Mrespectively. Then,
by [DB96, p. 94], we have
(2.10) g(R(X, Y )Z, P W ) = g(R(X, Y )Z, P W ) + B(X, Z)C(Y, P W )
−B(Y, Z )C(X, P W ),∀X, Y, Z, W ∈Γ(T M ).
In what follows, we shall make use of the following convention on the range
of indices: 1≤α, β, γ , µ, σ ≤n,0≤a, b, c ≤nand 0≤i, j, k ≤n+ 1.
3. Null mean curvature flow. The concept of null hypersurfaces (and
generally, submanifolds) can be presented alternatively by using a special
isometric immersion [DB96]. Let (M, g)be a 1-null manifold of dimension

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 5
n+ 1 and index q−1, with n > 0and q > 0. Let S(T M )be the screen
distribution of Mas in the previous section.
Suppose there exist a vector bundle Λof rank 1 over Msuch that T M =
T M ⊕Λis a semi-Riemannian vector bundle with a semi-Riemannian metric
gsatisfying
(C1)g(X, Y ) = g(X, Y ),g(Z, V ) = g(V, V 0)=0
for any X, Y ∈Γ(T M ),Z∈Γ(S(T M )) and V, V 0∈Γ(Λ). Since gis
non-degenerate on E, it follows that g(U, V )6= 0 for any non-zero vector
fields U∈Γ(Rad T M )and V∈Γ(Λ). Suppose there exists a torsion-free
linear connection ∇0on M and a linear connection ∇tr on Λsatisfying
(C2)g(∇0
XU, V ) + g(U, ∇tr
XV) = X(g(U, V ))
for any X∈Γ(T M ),U∈Γ(Rad T M)and V∈Γ(Λ). As
(3.1) T M =S(T M )⊥Rad T M,
we have the decompositions
∇0
XP Y =∇∗0
XP Y +h∗0 (X, P Y ),∀X, Y ∈Γ(T M ),(3.2)
∇0
XU=−A∗0
UX+∇∗tr
XU, ∀X∈Γ(T M ), U ∈Γ(Rad T M ),(3.3)
where Pis the projection morphism of T M onto S(T M ),{∇∗0
XP Y, A∗0
UX}
and {h∗0(X, P Y ),∇∗tr
XU}belong to S(T M )and Rad T M respectively. It fol-
lows that ∇∗0 and ∇∗tr are linear connections on the vector bundles S(T M)
and Rad T M , respectively. On the other hand, h∗0 and A∗0 are F(M)-bilinear
forms on Γ(T M )×Γ(S(T M )) and on Γ(Rad T M)×Γ(T M )respectively.
Moreover, we suppose that:
(C3)A∗0
UU= 0,g(A∗0
UX, Y ) = g(X, A∗0
UY),
(C4) (∇0
Xg)(P Y, P Z ) = (∇0
Xg)(U, U )=0,(∇0
Xg)(P Y, U ) = g(A∗0
UX, P Y ),
(C5) (∇0
XA∗0)(U, Y )=(∇0
YA∗0)(U, X ),
for all X, Y, Z ∈Γ(T M )and U∈Γ(Rad T M ), where (∇0
XA∗0)(U, Y ) =
∇0
XA∗0
UY−A∗0
∇∗tr
XUY−A∗0
U∇0
XY. Denote by R0the curvature tensor of ∇0
and further suppose that
(C6)g(R0(X, Y )Z, P W ) = g(A∗0
h∗0(X,P W )Y , Z)−g(A∗0
h∗0(Y ,P W )X, Z),
(C7)g(R0(X, Y )Z, V )=0for all X, Y, Z, W ∈Γ(T M )and V∈Γ(Λ).
Let (M, g, S(T M )) be a 1-null simply connected (n+1)-dimensional man-
ifold of index q−1, endowed with a vector bundle Λand geometric objects
g,∇0,∇tr,h∗0 and A∗0 satisfying conditions (C1)–(C7). Then there exists a
null isometric immersion F:Mn+1 →Mn+2 [DB96, Theorem 4.1] satisfying
g(X, Y ) = g(F∗X, F∗Y),∀X, Y ∈Γ(T M ),(3.4)

6F. MASSAMBA AND S. SSEKAJJA
and a vector bundle isomorphism F:tr(T M )→tr(T F (M)) such that
F∗(∇0
XY) = ∇F∗XF∗Y, F∗(A∗0
UX) = A∗
F∗UF∗X,(3.5)
F∗(h∗0(X, P Y )) = h∗(F∗X, F∗P Y ), F (∇tr
XV) = ∇t
XF V,(3.6)
for all X, Y ∈Γ(T M ),U∈Γ(Rad T M )and V∈Γ(tr(T M )), where
tr(T F (M)) is the null transversal vector bundle of F(M)with respect to
F∗S(T M ), and ∇,∇t,h∗,A∗are the geometric objects induced on F(M)
with respect to the immersion F. Certainly, F(M)is nothing but a 1-null
submanifold of Mn+2 and Fpreserves both the radical and screen distribu-
tion, that is, F∗maps respectively the radical subspace and the screen of the
domain to that of the base (see details in [DB96, p. 104]). More precisely,
Rad T F (M) = F∗Rad T M, S(T F (M)) = F∗S(T M ).(3.7)
Next, suppose that F(M)is a null hypersurface as described above. By
the method of [DB96], the null mean curvature vector Hof F(M)at p∈M
is a smooth vector field transversal to F(M)and given by
H= trs(h) = trs(B)N=SN,(3.8)
where N∈Γ(tr(FTM)) and S:= trs(B) = trs(A∗
E). The function Sis
called the null mean curvature of F(M). Thus, F(M)will be said to have
constant mean curvature if Sis constant.
In the following example, we calculate Sfor a null Monge hypersurface.
Example 3.1 (Null Monge hypersurface of Rn+2
1).Consider a non-zero
smooth function G:Σ→R, where Σis an open set in Rn+2
1. It is well-known
[DB96, DS10] that M={(x0, . . . , xn+1)∈Rn+2
1:x0=G(x1, . . . , xn+1)}
is a Monge hypersurface. Moreover, it is easy to check that Mis a null
hypersurface if and only if Gis a solution of the partial differential equation
Pn+1
i=1 (G0
xi)2= 1. Then Rad T M and tr(T M )are respectively spanned by
the global vector fields
E=∂
∂x0
+
n+1
X
i=1
G0
xi
∂
∂xi
and N=1
2−∂
∂x0
+
n+1
X
i=1
G0
xi
∂
∂xi.
The corresponding screen distribution is given by {Z1, . . . , Zn}, where Zα=
G0
xn+1
∂
∂xα−G0
xα
∂
∂xn+1 for α∈ {1, . . . , n}. Differentiating Pn+1
i=1 (G0
xi)2= 1
we get Pn+1
i=1 G0
xiG00
xixj= 0 for all i, j ∈ {1, . . . , n + 1}. Thus, ∇EE=
Pn+1
j=1 Pn+1
i=1 G0
xiG00
xixj
∂
∂xj= 0. Also, by simple calculations we have
∇ZαE=
n+1
X
i=1
(G0
xn+1 G00
xαxi−G0
xαG00
xn+1xi)∂
∂xi
.(3.9)

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 7
Since ∂
∂xβ= (G0
xn+1 )−1(Zβ−G0
xβ
∂
∂xn+1 ), (3.9) simplifies to
∇ZαE=
n
X
β=1
(G00
xαxβ−(G0
xn+1 )−1G0
xαG00
xn+1xβ)Zβ
(3.10)
+
n
X
β=1
(G00
xαxβG0
xβ−(G0
xn+1 )−1G0
xβG00
xn+1xβ)∂
∂xn+1
+ (G0
xn+1 G00
xαxn+1 −G0
xαG00
xn+1xn+1 )∂
∂xn+1
.
Using the fact that Pn
β=1(G00
xαxβ) + G0
xn+1 G00
xαxn+1 = 0 in (3.10) we deduce
∇ZαE=
n
X
β=1
(G00
xαxβ−(G0
xn+1 )−1G0
xαG00
xn+1xβ)Zβ.(3.11)
From (3.11) and Pn+1
i=1 G0
xiG00
xixj= 0 it follows that the mean curvature
function Sis given by
S=
n
X
α=1
(G00
xαxα−(G0
xn+1 )−1G0
xαG00
xn+1xα) =
n+1
X
i=1
G00
xixi.
3.1. Velocity on M.In order to introduce mean curvature flow for null
hypersurfaces we need to recall from [O83, pp. 10–11] some basic facts about
velocity in manifolds. Let σ:I→Rbe a smooth curve in a manifold M,
where Iis an open interval in R. In fact, Ihas a coordinate system consisting
of the identity map vof I. At each t∈R, one can picture the coordinate
vector (d/dv)(t)∈TtRas the unit vector at tin the positive v-direction.
Then, the velocity vector of σat t∈Iis given by
∂
∂t σ(t) = σ∗d
dv t∈Tσ(t)M,(3.12)
where σ∗denotes the differential of σ. Intuitively speaking, ∂
∂t σ(t)is the
vector rate of change of σat t∈I. Let x0, x1, . . . , xnbe a coordinate system
in Mat a point σ(t)of σ. Then we can express ∂
∂t σ(t)in coordinate form as
∂
∂t σ(t) =
n
X
a=0
d(xa◦σ)
dv (t)∂
∂xaσ(t)
.(3.13)
Moreover, if F:M→Mis a map, then Fcarries σto the curve F◦σ:
I→Min M. Furthermore, the differential map of Fpreserves velocities,
that is,
F∗∂
∂t σ(t)=∂
∂t (F◦σ)(t),∀t∈I.(3.14)

8F. MASSAMBA AND S. SSEKAJJA
A curve σis regular provided ∂
∂t σ(t)6= 0 for all t. If [w, y]is a closed interval
in R, then a curve segment σ: [w, y]→Mis a function that has a smooth
extension to an open interval containing [w, y]. Thus, ∂
∂t σ(t)is well-defined
even at the endpoints wand y. For other properties of the velocity vector
∂
∂t σ(t), see [O83].
Next, we adopt the concept of mean curvature flow for Riemannian hy-
persurfaces given in [H90]. For null hypersurfaces, we introduce the null mean
curvature flow (MCF) according to the following definition.
Definition 3.2.Let (Mn+1, g, S(T M )) be a compact null hypersur-
face of a semi-Riemannian manifold (Mn+2, g). Assume that F0:Mn+1 ×
[0, T )→Mn+2 smoothly immerses Mas a null hypersurface in Msuch
that conditions (C1)–(C7)and (3.4)–(3.7) hold. We say that M0:= F0(M)
evolves by its null mean curvature vector if there is a whole family F(·, t)
of smooth immersions with corresponding hypersurfaces Mt:= F(·, t)(M)
such that
∂F
∂t (p, t) = H(p, t), p ∈Mn+1, F (·,0) = F0,(3.15)
where H(p, t) := S(p, t)Nis the mean curvature vector of Mtat F(p, t).
It is important to notice that in the null MCF case, the flow of the surface
Mtis towards the transversal direction with a pseudo-speed equal to its null
mean curvature S:= trs(B) = trs(A∗
E), as opposed to the flow towards the
normal direction in the Riemannian (or semi-Riemannian) case. In fact, if σ:
[0, T )→Mis a smooth curve (also, σcan be seen as a particle in M) on an
interval [0, T )⊂R, then the action of the null isometric immersion Fon M,
that is, F:M→M, gives rise to a new curve (particle) F◦σ: [0, T )→M
in M. Then, for a given time t∈[0, T )and p=σ(t)∈M, the velocity vector
is given by (3.14) as ∂
∂t (F◦σ)(t), which is a vector in Mwith tangential
component ∂
∂t (F◦σ)(t)>and transversal component ∂
∂t (F◦σ)(t)t. If
at every t∈[0, T )we set ∂
∂t (F◦σ)(t)t=H=SN, then we obtain the
null MCF in the definition above. This choice makes sense due to the fact
that tangential velocity has no influence on the shape of the evolving null
hypersurface but it only controls the parametrization of the hypersurface
(see [M12, p. 10] for details).
Example 3.3.In what follows, we show how a double null foliation
of Minkowski spacetime evolves under null MCF. Let S0be an embedded
2-surface and Λ0,Λ0be the null hypersurfaces spanned by outgoing and
incoming null geodesics normal to S0. Let %:S0→Rbe a smooth function
(the null lapse, see [A13]) on S0, and E0be a null vector field normal to S0
(and tangential to Λ0). Also let N0be the null vector field normal to S0and

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 9
tangent to the null geodesics of Λ0such that
g(E0, N 0) = −%−2.(3.16)
Let us extend the fields E0,N0on Λ0,Λ0respectively so that ∇E0E0= 0
and ∇N0N0= 0. Also, extend %to a function on the null hypersurfaces Λ0
and Λ0, and consider the vector fields E00 =%2E0and N00 =%2N0. Define a
function uon Λ0by E00u= 1, with u= 0 on S0. Similarly, define a function
uon Λ0by N00u= 1, with u= 0 on S0.
Let S˜τbe the embedded 2-surface on Λ0such that u= ˜τ, and similarly,
let S˜τbe the embedded 2-surface on Λ0such that u= ˜τ. We also define N0
on Λ0so that N0is null and normal to S˜τand g(E0, N0) = −%−2.
Fig. 1. Double conical null foliation
Consider the affinely parametrized null geodesics which emanate from
the points on S˜τ, with initial tangent vector N0. These geodesics span null
hypersurfaces which we denote by C˜τ. Hence, Λ0∩C˜τ=S˜τ. Similarly,
we define E0globally (and the hypersurfaces C˜τsuch that their normal
is E0). Next, extend E00,N00 to global vector fields such that E00 =%2E0and
N00 =%2N0. Also, extend u, u to global functions such that
E00u= 0 and N00u= 0.(3.17)
Therefore, C˜τ={u= ˜τ}and C˜τ={u= ˜τ}, and hence u,uare optical
functions (see [A13] for more details). Moreover, u,usatisfy
∇u=−N0,∇u=−E0, E00 u= 1, N 00 u= 1,(3.18)
where ∇u,∇uare the gradients of u,urespectively. A proof of the above
relations can be found in [A13]. Observe from the above explanations that
we have foliated the spacetime with null hypersurfaces, seen as level surfaces
of uand u.

10 F. MASSAMBA AND S. SSEKAJJA
Next, we formulate our evolution model from the functions uand uas fol-
lows. From the previous explanations, we have Λ0={(x, u(x)) : x∈R3}=
Graph(u)⊂M4
1and Λ0={(x, u(x)) : x∈R3}= Graph(u)⊂M4
1. Then
we have the null immersions F(p, t)=(x(p, t), u(x(p, t), t)) and F(p, t) =
(x(p, t), u(x(p, t), t)) for Λ0, Λ0respectively. At this point, suppose that we
want to follow up the evolution of the future cone Λ0, starting from S0, along
its transversal direction. Then observe from Figure 1 that this is essentially
a flow along the past cone Λ0, as it is transversal to Λ0. Also, all points
of Λ0are moved along the transversal direction during this flow. By direct
calculations, we have
∂F
∂t =∂x
∂t ,∂u
∂xa
∂xa
∂t +∂u
∂t =∂x
∂t , g∇u, ∂x
∂t +∂u
∂t .(3.19)
From (3.17) and (3.18) we observe that the vector (N00, N00u) = (%2N0,0) =
(−%2∇u, 0) is null and always transversal to the future cone Λ0, and tangent
to the past cone Λ0. Thus, we can write (3.19), using the definition of null
MCF, as
∂x
∂t , g∇u, ∂x
∂t +∂u
∂t =S(−%2∇u, 0).(3.20)
Then from (3.20) we get
∂x
∂t =−S %2∇uand g∇u, ∂x
∂t +∂u
∂t = 0.(3.21)
From (3.21), (3.16) and (3.18) we deduce that
∂u
∂t =S%2g(∇u, ∇u) = −S %2%−2=−S .(3.22)
Finally, using (2.4), (3.21) and (3.22), we have ∂ x
∂t =−%2div(∇u)∇uand
∂u
∂t =−div(∇u). These PDE’s describe the flow of the above cones under
null MCF.
4. Short-time existence and evolution of geometric quantities.
For a Lorentzian ambient manifold, the null MCF equation (3.15) repre-
sents a strictly non-linear PDE. To show this, let (U;x0, . . . , xn)be local
coordinates around p∈ U ⊂ Mn+1 at t∈[0, T ), such that the local vector
fields X0=E=∂
∂x0, X1=∂
∂x1, . . . , Xn=∂
∂xnand N=∂
∂xn+1 span T M
at time t. Let us write F:= F(p, t),H:= H(p, t)and Ea:= F∗Xafor any
a∈ {0, . . . , n}. As Fis a null isometric immersion, [DB96, p. 109] shows that
Fis an affine immersion, that is, for any a, b ∈ {0, . . . , n}we have ∇EaEb=
F∗(∇XaXb) + B(Xa, Xb)N. Thus, since B(X0, X0) = B(E, E) = 0, the null

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 11
mean curvature vector Hcan be written, for all α, β , γ, µ ∈ {1, . . . , n}, as
H= (gαβ∇EαEβ)t=gαβ ∇EαEβ−(gαβ ∇EαEβ)>
(4.1)
=gαβ∇EαEβ−g(gαβ ∇EαEβ, Eγ)gγµ Eµ,
where (·)>and (·)tare the tangential and transversal components of (·)
respectively. Let Γa
αβ denote the ath component of ∇EαEβin the basis
{E0, . . . , En}; then (4.1) and (3.15) give ∂F
∂t =gαβ Γa
αβ(Ea−gγµ gaγ Eµ),
which is a system of non-linear PDE’s because (gαβ ), for α, β ∈ {1, . . . , n},
is a positive definite matrix which only depends on the first derivatives of F.
Moreover, using the local theory of parabolic PDE’s, the geometric equation
in (3.15) can be shown to be equivalent to a quasilinear scalar equation,
which may be solved for some short interval of time using linearization tech-
niques. This short-time solution may then be continued as long as it does
not become singular [T96].
Suppose that the null hypersurface (M, g)moves by null mean curva-
ture and let Fbe the smooth immersion in Definition 3.2. Notice that the
null immersion Fhas no well-defined d’Alembertian operator, but we can
compute its d’Alembertian-type operator ∆gFvia the d’Alembertians of
its coordinate functions with respect to bg(see [AET03]). Alternatively, we
can compute ∆gFfrom the Hessian Hessg(F)(regarded as a vector-valued
function on M0=F(M)). Let (M, g)be a null hypersurface of (M , g); then
∆gF= tr(Hessg(F)),(4.2)
where Hessg(F) = X(F∗Y)−F∗(∇0
XY)for all X, Y ∈Γ(T M ), and the trace
is taken on M0with respect to bg. In fact, using (3.5) we have
Hessg(F) = Xa(F∗Xb)−F∗(∇0
XaXb)(4.3)
=Xa(F∗Xb)−(∇F∗XaF∗Xb),∀, a, b ∈ {0, . . . , n}.
Contracting (4.3) with respect to bg, we get
tr(Hessg(F)) = g[ab](Xa(F∗Xb)−(∇F∗XaF∗Xb))(4.4)
=g[ab]B(F∗Xa, F∗Xb)N.
As Fis also an affine immersion [DB96, p. 109], (4.4) vanishes when a= 0
or b= 0 due to the fact that the local second fundamental form Bof a null
hypersurface is degenerate. Thus, from (4.4) and (4.1), we deduce that
∆gF= tr(Hessg(F)) = H.(4.5)
Now, from (4.5) we state the following.
Proposition 4.1.Let (M, g)be a null hypersurface in a Lorentzian man-
ifold (M , g)and F0:Mn+1 →Mn+2 a given immersion. Then the null MCF

12 F. MASSAMBA AND S. SSEKAJJA
equation in Definition 3.2 is given by
∂F
∂t =∆gF.(4.6)
Using Proposition 4.1 we can state and prove an existence theorem for
the null MCF in Definition 3.2 for a short time interval, for a compact null
hypersurface.
Theorem 4.2 (Short-time existence).Let (M, g)be a compact null hy-
persurface of a Lorentzian manifold (M , g)and F0:Mn+1 →Mn+2 a given
immersion. There exists a constant T > 0and a unique smooth family of
immersions F(·, t) : Mn+1 →Mn+1 such that
∂F
∂t (p, t) = H(p, t), F (·,0) = F0,(4.7)
where p∈Mand t∈[0, T ).
Proof. We start off by observing that F(·,0) = F0is an immersion,
so F(p, t)would be an immersion for some small t(see for instance [GP74,
p. 35]). Hence, we only focus on the existence of a solution of the PDE above.
Let us consider a vector field U=ua∂
∂xasuch that ∂˜
F
∂t =∆g˜
F+ua∂˜
F
∂xahas a
solution for the initial data F0,˜
F:M×[0, T )→Mn+1. We establish that the
same happens to the null mean curvature flow with initial data F0. Consider
a family ϕt:M→Mof diffeomorphisms of M. Set F(p) := ˜
F(ϕt(p), t),
where ˜
Fis as mentioned above. Then at a point p∈M, we have
∂F
∂t (p) = ∂˜
F
∂xa
(ϕt(p), t)∂ϕa
t
∂t (p) + ∂˜
F
∂t (ϕt(p), t)(4.8)
=∆g˜
F(ϕt(p), t) + ∂˜
F
∂xa
(ϕt(p), t)ua+∂ϕa
t
∂t (p).
Therefore, by (4.8), to get a solution to the null MCF equation it is enough
to find a family ϕtsuch that
∂ϕt
∂t =−U, ϕ0= id.(4.9)
Equation (4.9) is an initial value problem for a system of ODE’s, so we
can find a solution to it. Moreover, taking T > 0small enough we can
assume that ϕtis a diffeomorphism for any t∈[0, T ], the reason being
that the initial data is a diffeomorphism (the identity) and the fact that the
diffeomorphisms from a compact manifold into itself form a stable class (see
[GP74, p. 35]). Therefore, F(p) := ˜
F(ϕt(p), t)represents a solution of the null
MCF equation with initial data F0. In fact ∂F
∂t (p) = ∆g˜
F(ϕt(p), t) = ∆gF(p)
and F(p, 0) = ˜
F(ϕ0(p),0) = ˜
F(id(p),0) = F0(p).

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 13
Finally, we show that ∂˜
F
∂t =∆g˜
F+ua∂˜
F
∂xahas a solution. Consider the
vector Uwhose coordinates are ua=g[bc](Γa
bc −(Γ0)a
bc), where Γa
bc are the
coefficients of the induced connection on M, in which (Γ0)a
bc denotes the
coefficient at t= 0. Then by (4.4) we have
∂˜
F
∂t =∆g˜
F+ua∂˜
F
∂xa
=g[bc]∂2˜
F
∂xb∂xc
−(Γ0)a
bc
∂˜
F
∂xa,(4.10)
which is a system of quasilinear parabolic PDE’s because (g[bc])is a positive-
definite matrix which only depends on the first derivatives of ˜
F. Hence the
local theory of parabolic PDE’s (see, for instance, [T96]) and the fact that
Mis compact give us the existence and uniqueness of the solution in a short
time interval [0, T ).
Now, we discuss the extrinsic geometry of Mt. To that end set Ea:=
F∗Xa=∂F
∂xa,τa:= τ(Ea),Bαβ =B(Eα, Eβ),Cαβ =C(Eα, Eβ)and ˜
S:=
Pn
α=1 C(Eα, Eα).
Theorem 4.3.Let (M, g)be a screen integrable null hypersurface of a
Lorentzian manifold (M , g). Let F0:M→Mbe a given immersion. On any
solution Mtof the null MCF (3.15), the induced objects g,B,C,S, and the
null normal Eevolve as follows:
∂
∂t gαβ =−2SCαβ,(4.11)
∂
∂t gαβ = 2SCαβ,(4.12)
∂
∂t E=−∇sS − S gαβ ταEβ,(4.13)
∂
∂t Bαβ = Hesss(S)− SB∂
∂xα
, AN
∂
∂xβ+∂
∂xβ
(S)τα
(4.14)
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ− SΓ∗γ
αβτγ,
∂
∂t S=∆S+SH +gαβ∂
∂xβ
(S)τα+∂
∂xα
(S)τβ+S∂
∂xα
(τβ)(4.15)
+Sτατβ− SΓ∗γ
αβτγ),
where H:= CαβBαβ = trs(A∗
E◦AN), for any α, β, γ ∈ {1, . . . , n}.
Proof. By virtue of (3.4), the null MCF equation (3.15) and the local
symmetry of the Levi-Civita connection, we derive

14 F. MASSAMBA AND S. SSEKAJJA
∂
∂t gαβ =∂
∂t g(Eα, Eβ) = g∂
∂t Eα, Eβ+gEα,∂
∂t Eβ
(4.16)
=g∂
∂xα
H, Eβ+gEα,∂
∂xβ
H.
As Fpreserves both the radical and screen subspaces (see [DB96, p. 104]),
we can see, by (3.7), that Eαbelongs to the screen distribution in Mtfor all
α∈ {1, . . . , n}. From the above facts and as the mean curvature vector H
is transversal to Mt, we get g(Eα,H)=0. Hence, from the Gauss–Codazzi
equations (2.1), (2.3), we get
g∂
∂xα
H, Eβ=−gH,∂
∂xα
Eβ=−SCαβ.(4.17)
Then (4.11) follows from (4.16) and (4.17) and the symmetry of Cin α
and β.
Relation (4.12) follows easily from (4.11) by the product rule.
Next, we prove (4.13). Decomposing the vector field ∂
∂t Ein the basis
{E1, . . . , En}and observing g(E, Ea)=0for all a∈ {0, . . . , n}, we have
∂
∂t E=gαβg∂
∂t E, EαEβ
(4.18)
=gαβ∂
∂t g(E, Eα)−gE, ∂
∂t EαEβ
=−gαβ∂
∂xα
g(E, H)−g∂
∂xα
E, HEβ,
in which we have used the symmetry of the Levi-Civita connection. But
g(E, H) = g(E , SN) = Sg(E, N ) = Sand by (2.1) and (2.4) the second
term in parenthesis in (4.18) simplifies as
g∂
∂xα
E, H=Sg∂
∂xα
E, N =−Sτα.(4.19)
Inserting (4.19) back into (4.18), we get
∂
∂t E=−gαβ ∂
∂xα
(S)Eβ=−∇sS − S gαβ ταEβ,
which proves (4.13).
From the Gauss formula (2.1) we derive
∂
∂t Bαβ =∂
∂t g∂
∂xα
Eβ, E
(4.20)
=g∂2
∂t∂xα
Eβ, E+g∂
∂xα
Eβ,∂
∂t E.

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 15
As we are working in local coordinates and as R= 0 (this follows easily from
the Gauss–Codazzi equations and (C7); see details in [DB96, p. 103]), one
gets ∂2
∂t∂ xαEβ=∂2
∂xα∂ t Eβ=∂2
∂xα∂ xβ
Hand so (4.20) gives
∂
∂t Bαβ =g∂2
∂xα∂xβ
H, E+g∂
∂xα
Eβ,∂
∂t E.(4.21)
Now, using the Gauss–Codazzi relations (2.1)–(2.2) together with ∂
∂xβH=
∂
∂xβ(SN) = ∂
∂xβSN+S∂
∂xβNgives
∂
∂t Bαβ =g∂
∂xα ∂
∂xβ
(S)N, E− S B∂
∂xα
, AN
∂
∂xβ
(4.22)
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ+g∂
∂xα
Eβ,∂
∂t E
=∂
∂xα∂
∂xβ
(S)− SB∂
∂xα
, AN
∂
∂xβ+∂
∂xβ
(S)τα
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ+g∂
∂xα
Eβ,∂
∂t E.
Then applying (4.13) to (4.22) we get
(4.23) ∂
∂t Bαβ =∂
∂xα∂
∂xβ
(S)− SB∂
∂xα
, AN
∂
∂xβ+∂
∂xβ
(S)τα
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ
−gγµ g∂
∂xα
Eβ,∂
∂xγ
(S)Eµ− Sgγµ τγg∂
∂xα
Eβ, Eµ
=∂
∂xα∂
∂xβ
(S)− SB∂
∂xα
, AN
∂
∂xβ+∂
∂xβ
(S)τα
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ−Γ∗γ
αβ
∂
∂xγ
(S)− SΓ∗γ
αβτγ,
where Γ∗γ
αβ are the coefficients of the metric connection ∇∗. Then from (4.23)
we deduce that
∂
∂t Bαβ = Hesss(S)− SB∂
∂xα
, AN
∂
∂xβ+∂
∂xβ
(S)τα
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ− SΓ∗γ
αβτγ,
where Hesss(S) := ∂
∂xα∂
∂xβ(S)−Γγ
∗αβ
∂
∂xγ(S), which proves (4.14).

16 F. MASSAMBA AND S. SSEKAJJA
Finally, we infer from (4.12) and (4.14) that
∂
∂t S=∂
∂t gαβBαβ +gαβ ∂
∂t Bαβ
= 2SCαβBαβ +gαβ Hesss(S)− Sg(ANEβ, A∗
EEα) + ∂
∂xβ
(S)τα
+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ− SΓ∗γ
αβτγ
=∆sS+ 2SCαβBαβ − S tr(A∗
E◦AN)
+gαβ∂
∂xβ
(S)τα+∂
∂xα
(S)τβ+S∂
∂xα
(τβ) + Sτατβ− SΓ∗γ
αβτγ),
which proves (4.15).
Next, we will derive the evolution equation for the squared norm (with
respect to the screen distribution) |A∗
E|2
sof the screen shape operator A∗
E.
To that end, let us consider the orthonormal basis {X1, . . . , Xn}, where
Xα=∂
∂xα, of S(T M )around a point p∈Msuch that the induced Levi-
Civita connection ∇∗of S(T M )satisfies (∇∗
XαXβ)(p)=0 and C(X0, Xα)=0.
Observe that the above assumptions implies that locally the curvature R∗
of S(T M )vanishes. Furthermore, we will assume that the 1-form τvanishes
on S(T M ). As an example we have the following.
Example 4.4 (Null cone of Rn+2
1).Let Rn+2
1be the space Rn+2 endowed
with a semi-Euclidean metric
g(x, y) = −x0y0+
n+1
X
a=0
xayax=
n+1
X
A=0
xA∂xA,
where ∂xA:= ∂
∂xA. Then the null cone Λn+1
0is given by the equation x2
0=
Pn+1
a=1 x2
a,x06= 0. It is well-known (see for example [DB96, DS10]) that Λn+1
0
is a null hypersurface of Rn+2
1, in which the radical distribution is spanned
by a global vector field E=Pn+1
A=0 xA∂xAon Λn+1
0. The transversal bundle
is spanned by a global section Ngiven by N=1
2x2
0
{−x0∂x0+Pn+1
a=1 xa∂xa}.
Moreover, Ebeing the position vector field, one gets ∇XE=∇XE=Xfor
any X∈Γ(T M ). Consequently, A∗
EX+τ(X)E+X= 0. Noticing that the
operator A∗
Eis screen-valued, we infer from the last relation that
A∗
EX=−P X, τ (X) = −g(X, N ) = −λ(X)(4.24)
for any X∈Γ(T M ). Next, any X∈Γ(S(T Λn+1
0)) can be expressed as
X=Pn+1
a=1 ˜
Xa∂xa, where {˜
X1,..., ˜
Xn+1}satisfy Pn+1
a=1 xa˜
Xa= 0. From
the second relation of (4.24) we clearly see that τ(X)=0for any X∈
Γ(S(T M )).

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 17
Next, we turn our attention to the evolution equation of |A∗
E|2
s. To that
end, we will need the following result.
Proposition 4.5.Let (M, g)be a screen integrable null hypersurface of
a Lorentzian manifold (M , g). The local second fundamental form Bsatisfies
∆sBαβ = Hesss(S)− HBαβ +SC(Xα, A∗
EXβ)
for all α, β ∈ {1, . . . , n}, where H:= trs(A∗
E◦AN).
Proof. By the assumption R= 0, the Gauss–Codazzi equation (3.1) of
null hypersurfaces given in [DB96, p. 93] implies
(∇Xh)(Y, Z )=(∇Yh)(X, Z)(4.25)
for all X, Y, Z ∈Γ(T M ), where
(∇Xh)(Y, Z ) = ∇t
Xh(Y, Z )−h(∇XY, Z)−h(Y, ∇XZ).(4.26)
As ∇∗
XαXβ= 0, we have
h(∇XαXβ, Xγ) = h(∇∗
XαXβ, Xγ) + C(Xα, Xβ)h(E, Xγ) = 0,
in which we have taken into account (2.3) and the fact h(E, X)=0for any
X∈Γ(T M ). Thus, (4.25) and (4.26) imply
∇t
Xγh(Xα, Xβ) = ∇t
Xαh(Xγ, Xβ).(4.27)
Differentiating (4.27) and applying the definition of curvature, we get
(4.28) ∇t
Xµ∇t
Xγh(Xα, Xβ) = ∇t
Xµ∇t
Xαh(Xγ, Xβ)
=∇t
Xα∇t
Xµh(Xγ, Xβ)−(R(Xα, Xµ)h)(Xβ, Xγ),
where
(4.29) (R(Xα, Xµ)h)(Xβ, Xγ) = Rt(Xα, Xµ)h(Xβ, Xγ)
−h(R(Xα, Xµ)Xβ, Xγ)−h(Xβ, R(Xα, Xµ)Xγ),
and Rtis the curvature tensor of the transversal bundle, given by
Rt(X, Y )N=∇t
X∇t
YN− ∇t
Y∇t
XN− ∇t
[X,Y ]N(4.30)
for any X, Y ∈Γ(T M )and N∈Γ(tr(T M )). Then applying (4.28)–(4.30)
to ∇t
XN=τ(X)N, we get
∇t
Xµ∇t
Xγh(Xα, Xβ) = ∇t
Xα∇t
Xµh(Xβ, Xγ) + h(R(Xα, Xµ)Xβ, Xγ)(4.31)
+h(Xβ, R(Xα, Xµ)Xγ)
=∇t
Xα∇t
Xβh(Xµ, Xγ) + h(R(Xα, Xµ)Xβ, Xγ)
+h(Xβ, R(Xα, Xµ)Xγ),
where in the last equality we have used (4.27). Now, as h(X, Y ) = B(X, Y )N
and B(X, Y ) = g(A∗
EX, Y )for any X, Y ∈Γ(T M ), and because of the

18 F. MASSAMBA AND S. SSEKAJJA
assumption τ= 0 on the screen distribution, (4.31) reduces to
Xµ(Xγ(Bαβ)) = Xα(Xβ(Bµγ )) + B(R(Xα, Xµ)Xβ, Xγ)(4.32)
+B(Xβ, R(Xα, Xµ)Xγ)
=Xα(Xβ(Bµγ )) + g(A∗
EXγ, R(Xα, Xµ)Xβ)
+g(A∗
EXβ, R(Xα, Xµ)Xγ).
Next, as R= 0, from the Gauss–Codazzi relation (2.10) we get
(4.33) g(R(X, Y )Z, P W ) = B(Y , Z)C(X, P W )−B(X, Z)C(Y, P W )
for any X, Y, Z, W ∈Γ(T M ). Applying (4.33) to (4.32) reduces it to
Xµ(Xγ(Bαβ)) = Xα(Xβ(Bµγ )) + Bµβ C(Xα, A∗
EXγ)(4.34)
−BαβC(Xµ, A∗
EXγ) + Bµγ C(Xα, A∗
EXβ)
−Bαγ C(Xµ, A∗
EXβ).
We are working locally around a point p∈M, so τ([Xα, Xβ]) = 0. Taking
this into account, together with the assumption τ(Xα)=0for all α∈
{1, . . . , n}, and R= 0, we deduce from the Gauss–Codazzi equation [DB96,
(3.12), p. 95] that
C(Xα, A∗
EXγ) = C(Xγ, A∗
EXα).(4.35)
Finally, inserting (4.35) in (4.34) and taking trace with respect to µand γ,
we obtain the desired result.
Theorem 4.6.Let (M, g)be a screen integrable null hypersurface of a
Lorentzian manifold (M , g). Under null MCF, the squared norm |A∗
E|2
sof the
screen shape operator A∗
Eevolves according to
∂|A∗
E|2
s
∂t =∆s|A∗
E|2
s+ 2H|A∗
E|2
s−2|∇sB|2
s,
where H:= trs(A∗
E◦AN).
Proof. Consider gαβ(p) = δαβ . Then |A∗
E|2
s=gαµgγ β BαµBγ β =Bαβ Bαβ .
Using this relation and Proposition 4.5 we have
(4.36) ∆s|A∗
E|2
s= 2Bαβ∆sBαβ + 2|∇sB|2
s
= 2BαβHesss(S)−2H|A∗
E|2
s+ 2|∇sB|2
s+ 2Strs(A∗2
E◦AN).
On the other hand, we can compute the time derivative of |A∗
E|2
sfrom |A∗
E|2
s=
gαβgγµ Bαγ Bβµ and the evolution equations of gand Bin Theorem 4.3 with
τ= 0 on the screen distribution as follows:
∂|A∗
E|2
s
∂t = 2SCαβ gγ µBαγ Bβµ + 2Sgαβ Cγ µBαγ Bβ µ
(4.37)
+gαβgγµ {Hesss(S)− SB(Xα, ANXγ)}Bβµ
+gαβgγµ Bαγ {Hesss(S)− SB(Xβ, ANXµ)}.

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 19
Rearranging some indices in (4.37), we get
∂|A∗
E|2
s
∂t = 4SCαβ gγ µBαγ Bβµ + 2gαβ gγ µ{Hesss(S)− S B(Xα, ANXγ)}Bβ µ
(4.38)
= 4Strs(A∗2
E◦AN)+2Bαγ Hesss(S)−2Strs(A∗2
E◦AN)
= 2Strs(A∗2
E◦AN)+2Bαγ Hesss(S).
Putting together (4.36) and (4.38) we get the desired result.
In order to discuss some geometric implications of the evolution equations
in Theorems 4.3 and 4.6, we need the following well-known concept in null
geometry.
A null hypersurface (M, g)of a semi-Riemannian manifold (M , g)is
called screen conformal [DB96, p. 51] if there exist a non-vanishing smooth
function ψon a neighborhood Uin Msuch that AN=ψA∗
E, or equivalently
C(X, P Y ) = ψB(X, Y ),∀X, Y ∈Γ(T M ).(4.39)
We say that Mis screen homothetic if ψis a constant function on M.
Example 4.7.Consider the null cone in Example 4.4. By straightforward
calculation, one gets g(∇EX, E) = −Pn+1
a=1 xaXa= 0, which implies that
∇EX∈Γ(S(T Λn+1
0)). Hence, ANE= 0. Using Gauss–Codazzi equations,
we calculate C(X, Y ) = g(∇XY , N) = g(∇XY , N) = −1
2x2
0
g(X, Y )for any
X, Y ∈Γ(S(T Λn+1
0)). Consequently,
ANX=−1
2x2
0
P X, ∀X, Y ∈Γ(S(T Λn+1
0)).(4.40)
Combining (4.24) and (4.40) we deduce that
ANX=1
2x2
0
A∗
EX, ∀X∈Γ(S(T Λn+1
0)).(4.41)
Hence, Λn+1
0is a screen globally conformal null hypersurface of Rn+2
1, with
a positive conformal factor ψ=1
2x2
0
globally defined on Λn+1
0.
Using the previous concept and Theorem 4.6 we state the following.
Corollary 4.8.Under the hypothesis of Theorem 4.6, if the initial null
hypersurface M0is screen conformal and the 1-form τvanishes on S(T M0),
then
∂|A∗
E|2
s
∂t =∆s|A∗
E|2
s+ 2ψ|A∗
E|4
s−2|∇sB|2
s.
From Corollary 4.8 we notice that the quadratic term in |A∗
E|2
swill cause
finite time blow-up in the evolution equation of |A∗
E|2
s. To understand the
long-term behavior of solutions near such blow-ups (or singularities) requires

20 F. MASSAMBA AND S. SSEKAJJA
one to obtain a priori estimates. Such estimates can be integral or pointwise.
In the latter case, the following maximum principle is always used.
Theorem 4.9 ([M12]).Assume that g(t), for t∈[0, T ), is a family of
Riemannian metrics on a manifold M, with a possible boundary ∂M , such
that the dependence on tis smooth. Let u:M×[0, T )→Rbe a smooth
function satisfying
∂u
∂t ≤∆g(t)u+g(X(p, u, ∇u, t),∇u)g(t)+f(u),(4.42)
where Xand fare respectively a continuous vector field and a locally Lip-
schitz function in their arguments. Suppose that for every t∈[0, T )there
exists a value δ > 0and a compact subset K ⊂ M\∂M such that at every
time t0∈(t−δ, t +δ)∩[0, T 0)the maximum of u(·, t0)is attained at least
at one point of K(this is clearly true if Mis compact without boundary).
Set umax(t) := maxp∈Mu(p, t). Then the function umax is locally Lipschitz,
hence differentiable at almost every t∈[0, T ), and at every differentiability
time we have
dumax(t)
dt ≤f(umax(t)).(4.43)
Consequently, if v: [0, T 0)→Ris a solution of the ODE
dv(t)
dt =f(v(t)), v(0) = umax(0),(4.44)
for T0≤T, then u≤vin M×[0, T 0). Moreover, if Mis connected and at
some time ˜
t∈(0, T 0)we have umax(˜
t) = v(˜
t), then u=vin M×[0,˜
t], that
is, u(·, t)is constant in space.
Notice that analogous results hold for the minimum of the solution of the
opposite partial differential inequality. Moreover, the maximum principle for
elliptic equations easily follows as the special case where all the quantities
around do not depend on the time variable t. We will apply the above maxi-
mum principle to some of evolution equations derived earlier, when the initial
null hypersurface M0is compact and mean convex. The hypersurface M0is
mean convex if S ≥ 0everywhere. It is well-known that mean convexity is
preserved by the mean curvature flow [M12].
Theorem 4.10.Let M0be a mean convex, screen conformal null hyper-
surface. If |A∗
E|2
sis not bounded as t→T < ∞during the null MCF of a
compact null hypersurface, then it must satisfy the following lower bound for
its blow-up rate:
max
p∈M|A∗
E|2
s(p, t)≥1
2ψ(T−t)

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 21
for every t∈[0, T ). Hence,
lim
t→Tmax
p∈M|A∗
E|2
s(p, t) = ∞.
Proof. From Corollary 4.8 and the maximum principle (Theorem 4.9),
we deduce that
∂|A∗
E|2
smax
∂t ≤2ψ|A∗
E|4
smax.
Notice that when the ambient space Mis Lorentzian, then the screen dis-
tribution of M0is Riemannian. Consequently, |A∗
E|2
smax is always positive,
otherwise we would get the case A∗
E= 0 (that is, M0is totally geodesic
in M), rendering M0a hyperplane in M, thereby contradicting the com-
pactness assumption on M0.
More precisely, there are no compact hypersurfaces with zero mean curva-
tures [M12]. Therefore, we can divide both sides by |A∗
E|4
smax obtaining the
following differential inequality for the locally Lipschitz function 1
|A∗
E|2
smax ,
holding at almost every t∈[0, T ):
−d
dt
1
|A∗
E|2
smax
≤2ψ.
Integration of the above inequality in [t, t0]⊂[0, T )gives
1
|A∗
E(·, t)|2
smax
−1
|A∗
E(·, t0)|2
smax
≤2ψ(t0−t).
Suppose that A∗
Eis not bounded in [0, T ), that is, there exists a sequence of
times t0
i→Tsuch that |A∗
E(·, t0
i)|2
smax → ∞. Considering these times t0
iin
the above inequality and letting i→ ∞, we get
1
|A∗
E(·, t)|2
smax
≤2ψ(T−t).
The above result shows that there will also be singularities in null MCF if
the initial null hypersurface M0is screen conformal since the quantity |A∗
E|2
s
blows up in finite time. When this happens we say that Tis a singular time
for the null MCF. Moreover, we have the following.
Theorem 4.11.Given a compact, immersed screen conformal null hy-
persurface M0in M, there exists a unique null MCF defined on a maximal
interval [0, Tmax). Moreover, Tmax is given by
max
p∈M|A∗
E|2
s≥1
2ψ(Tmax −t).
Let us turn to the evolution equation of S. From Theorem 4.3 with τ= 0
on the screen distribution, we have
∂S
∂t =∆sS+S H,where H:= trs(A∗
E◦AN).(4.45)

22 F. MASSAMBA AND S. SSEKAJJA
Theorem 4.12.Let M0be a mean convex and compact screen confor-
mal null hypersurface of M. Under null MCF, the minimum Smin of Sis
increasing, hence Sis positive for every positive time.
Proof. Consider the interval (t0, t1)⊂R+and suppose for contradiction
that Smin(t)<0and Smin(t0)=0for any tin this interval. Assume that
|A∗
E|2
sis bounded on (t0, t1), that is, |A∗
E|2
s≤A. Then relation (4.45) and
the maximum principle imply that
∂Smin
∂t ≥ψASmin,∀t∈(t0, t1).(4.46)
Integrating (4.46) over [s, t]⊂(t0, t1)we get
Smin(t)≥expt

s
ψASmin(s).(4.47)
Then, letting s→t+
0in (4.47) gives Smin(t)≥0for all t∈(t0, t1), which is
a contradiction. Next, let k1, . . . , knbe the eigenvalues of A∗
Ewith respect
to {X1, . . . , Xn}. It is easy to show that
|A∗
E|2
s−1
nS2=1
nX
1≤α≤β≤n
(kβ−kα)2,(4.48)
from which we deduce |A∗
E|2
s−1
nS2≥0. Using this inequality and (4.45) we
get
∂S
∂t =∆S+S H ≥ ∆S+ψ
nS3.(4.49)
Let u=−S,X= 0 and f(x) = ψ
nx3. Then if Smin(0) = 0, the ODE has
a solution v(t) = 0. Hence, if for some time ˜
twe have Smin(˜
t) = 0, then
Smin(·,˜
t)is constant on Mtand equal to zero. But there are no compact
hypersurfaces with zero mean curvature (see [M12]). Hence, Smin is always
increasing during the flow, and Sis positive at every positive time.
Next, suppose that M0admits a symmetric induced Ricci tensor. Denote
by ˜
Rthe corresponding scalar curvature. As R= 0, we deduce from of [DS10,
p. 69, (2.4.12)] that
˜
R= trs(A∗
E) trs(AN)−trs(A∗
E◦AN).(4.50)
Theorem 4.13.If the scalar curvature ˜
Ris positive and bounded on a
screen homothetic initial null hypersurface M0with ψ > 0, then it remains
bounded for all positive times t.
Proof. As M0is screen homothetic, (4.50) gives
˜
R=ψS2−ψ|A∗
E|2
s,(4.51)
where ψis a constant function on M0. Differentiating (4.51) and using the
evolution equations of Sand |A∗
E|2
s(see Theorems 4.3, 4.6 and Corollary 4.8),

NULL HYPERSURFACES EVOLVED BY THEIR MEAN CURVATURE 23
we get
∂˜
R
∂t = 2ψS(∆sS+ψS |A∗
E|2
s)−ψ(∆s|A∗
E|2
s+ 2ψ|A∗
E|4
s−2|∇sB|2
s)(4.52)
=∆s(ψS2)−∆s(ψ|A∗
E|2
s)+2ψ2S2|A∗
E|2
s−2ψ2|A∗
E|4
s
−2ψ|∇sS|2
s+ 2ψ|∇sB|2
s
=∆s˜
R+ 2ψ|A∗
E|2
s˜
R+ 2ψ(|∇sB|2
s− |∇sS|2
s).
Observe that the term |∇sB|2
s−|∇sS|2
sis non-positive and hence (4.52) gives
∂˜
R
∂t ≤∆s˜
R+ 2ψ|A∗
E|2
s˜
R.(4.53)
Thus, taking T0< T , if ωis the maximum of |A∗
E|2
son M×[0, T 0], we deduce
from (4.53) that ∂˜
R
∂t ≤∆s˜
R+2ψω ˜
Ron M×[0, T 0]. By the maximum principle
(Theorem 4.9), we have d˜
Rmax
dt ≤2ψω|˜
R|max, from which the result follows
by integration and the arbitrariness of T0.
Acknowledgements. This work is based on the research supported
wholly/in part by the National Research Foundation of South Africa (grant
numbers: 95931 and 106072).
REFERENCES
[A97] B. Andrews, Monotone quantities and unique limits for evolving convex hyper-
surfaces, Int. Math. Res. Notices 1997, no. 20, 1001–1031.
[A13] S. Aretakis, Lecture Notes on General Relativity, Princeton Univ., 2013.
[AK03] A. Ashtekar and B. Krishnan, Dynamical horizons and their properties, Phys.
Rev. D (3) 68 (2003), no. 10, 104030, 25 pp.
[A09] C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Appl. Sci. 11 (2009),
9–18.
[AET03] C. Atindogbé, J.-P. Ezin and J. Tossa, Pseudoinversion of degenerate metrics,
Int. J. Math. Math. Sci. 2003, no. 55, 3479–3501.
[DL14] J. Dong and X. Liu, Totally umbilical lightlike hypersurfaces in Robertson–Walker
spacetimes, ISRN Geom. 2014, 974695, 10 pp.
[DB96] K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Man-
ifolds and Applications, Math. Appl. 364, Kluwer, 1996.
[DS10] K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds,
Frontiers Math., Birkhäuser, 2010.
[GH86] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,
J. Differential Geom. 23 (1986), 69–96.
[GP74] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.
[GO16] M. Gutiérrez and B. Olea, Induced Riemannian structures on null hypersurfaces,
Math. Nachr. 289 (2016), 1219–1236.
[H84] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differ-
ential Geom. 20 (1984), 237–266.

24 F. MASSAMBA AND S. SSEKAJJA
[H90] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow,
J. Differential Geom. 31 (1990), 285–299.
[H98] G. Huisken, Evolution of hypersurfaces by their curvature in Riemannian man-
ifolds, in: Proc. Int. Congress of Mathematicians (Berlin, 1998), Vol. II, Doc.
Math. 1998, Extra Vol. II, 349–360.
[J13] D. H. Jin, Ascreen lightlike hypersurfaces of an indefinite Sasakian manifold,
J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 20 (2013), 25–35.
[K81] K. Katsuno, Null hypersurfaces in Lorentzian manifolds II, Math. Proc. Cam-
bridge Philos. Soc. 89 (1981), 525–532.
[K96] D. N. Kupeli, Singular Semi-Riemannian Geometry, Math. Appl. 366, Kluwer,
1996.
[M12] C. Mantegazza, Lecture Notes on Mean Curvature Flow, Progr. Math. 290,
Birkhäuser, 2012.
[MS16] F. Massamba and S. Ssekajja, Quasi generalized CR-lightlike submanifolds of
indefinite nearly Sasakian manifolds, Arab. J. Math. (Springer) 5 (2016), 87–101.
[O83] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure
Appl. Math. 103, Academic Press, 1983.
[T96] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Appl.
Math. Sci. 117, Springer, 1996.
Fortuné Massamba, Samuel Ssekajja
School of Mathematics, Statistics and Computer Science
University of KwaZulu-Natal
Private Bag X01
Scottsville 3209, South Africa
E-mail: massfort@yahoo.fr
Massamba@ukzn.ac.za
ssekajja.samuel.buwaga@aims-senegal.org