Lev Buhovsky's research while affiliated with Tel Aviv University and other places
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Publications (39)
An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count based on the theory of persistence modules origi...
On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite dimensional Fr\'echet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pullbacks. We show that this action is flexible: for a...
We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a ``large enough" tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact type boundaries, or when the amb...
Courant's theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We sh...
We prove that on a symplectic sphere, the group of Hamiltonian homeomorphisms in the sense of Oh and M\"uller is a proper normal subgroup of the group of finite energy Hamiltonian homeomorphisms. Moreover we detect infinite-dimensional flats inside the quotient of these groups endowed with the natural Hofer pseudo-metric.
We prove a quantitative h-principle statement for subcritical isotropic embeddings. As an application, we construct a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least 6.
In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous...
We present a short proof of the Fabry quotient theorem, which states that for a complex power series with unit radius of convergence, if the quotient of its consecutive coefficients tends to s, then the point z=s is a singular point of the series. This proof only uses material from undergraduate university studies.
Our first main result states that the spectral norm \(\gamma \) on \( \mathrm {Ham}(M, \omega ) \), introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the \(C^0\) topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent resu...
We prove a quantitative $h$-principle statement for subcritical isotropic embeddings. As an application, we construct a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least $6$.
We present a short proof of the Fabry quotient theorem. This proof only uses material from undergraduate university studies.
In this article we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous s...
We construct a Riemannian metric on the 2D torus, such that for infinitely many eigenvalues of the Laplace–Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of our construction implies that each of these eigenfunctions has a level set with infinitely many connected components (i.e.,...
We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points.
Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of K...
For any strictly convex planar domain $\Omega \subset \mathbb{R}^2$ with a $C^\infty$ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi-Merlose. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is...
An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at the origin $|u|\le 1$ on a $(1-\varepsilon)$ portion of $Q$...
The Poisson bracket invariant of a cover of a closed symplectic manifold measures how much a collection of smooth functions forming a partition of unity subordinate to the cover, can become close to being Poisson commuting. We introduce a new approach to this invariant in dimension 2, which enables us to significantly improve previously known lower...
We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof...
A set $A$ in a finite dimensional Euclidean space is \emph{monovex} if for every two points $x,y \in A$ there is a continuous path within the set that connects $x$ and $y$ and is monotone (nonincreasing or nondecreasing) in each coordinate. We prove that every open monovex set as well as every closed monovex set is contractible, and provide an exam...
The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal o...
We prove here a quantitative $h$-principle statement that applies to isotropic embeddings of discs. We then apply it to get $C^0$-flexibility and rigidity results in symplectic geometry. On the flexible side, we prove that a symplectic homeomorphism might take a symplectic disc to a smooth isotropic one. We also get a $C^0$-rigidity result for the...
This paper proceeds with the study of the (Formula presented.)-symplectic geometry of smooth submanifolds, as initiated in Humilière et al. (Duke Math J 164(4), 767–799, 2015) and Opshtein (Ann Sci Éc Norm Supér 42(5), 857–864, 2009), with the main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacit...
This paper studies the action of symplectic homeomorphisms on smooth
submanifolds, with a main focus on the behaviour of symplectic homeomorphisms
with respect to numerical invariants like capacities. Our main result is that a
symplectic homeomorphism may preserve and squeeze codimension $4$ symplectic
submanifolds ($C^0$-flexibility), while this i...
In this note, we generalise a result of Lalonde, McDuff and Polterovich
concerning the $ C^0 $ flux conjecture, thus confirming the conjecture in new
cases of a symplectic manifold. Also, we prove the continuity of the flux
homomorphism on the space of smooth symplectic isotopies endowed with the $ C^0
$ topology, which implies the $ C^0 $ rigidity...
On any closed symplectic manifold of dimension greater than 2, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to 1, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast wit...
Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we
consider all Riemannian metrics on M, which are compatible with the symplectic
structure \omega. For each such metric, we look at the first eigenvalue
\lambda_1 of the Laplacian associated with it. We show that \lambda_1 can be
made arbitrarily large, when we vary the met...
We introduce new invariants associated to collections of compact subsets of a
symplectic manifold. They are defined through an elementary-looking variational
problem involving Poisson brackets. The proof of the non-triviality of these
invariants involves various flavors of Floer theory. We present applications to
approximation theory on symplectic...
We study the class of pseudo-norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such pseudo-norm that is continuous with respect to the C
∞-topology, is dominated from above by the L
∞-norm. As a corollary, we obtain...
We prove that a topological Hamiltonian flow as defined by Oh and Muller, has
a unique $L^{(1,\infty)}$ generating topological Hamiltonian function. This
answers a question raised by Oh and Muller, and improves a previous result of
Viterbo.
In this paper we present an explicit construction of a relative symplectic packing. This confirms the sharpness of the upper bound for the relative packing of a ball into the pair (CP2, TCliff2) of the standard complex projective plane and the Clifford torus, obtained by Biran and Cornea.
In this paper we present an explicit construction of a relative symplectic packing. This confirms the sharpness of the upper bound for the relative packing of a ball into the pair (CP^2, T^2) of the standard complex projective plane and the Clifford torus, obtained by Biran and Cornea.
In this paper we introduce a new method for approaching the C
0-rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C
0 perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous resul...
Given a diffeomorphism of the interval, consider the uniform norm of the
derivative of its n-th iteration. We get a sequence of real numbers called the
growth sequence. Its asymptotic behavior is an invariant which naturally
appears both in smooth dynamics and in geometry of the diffeomorphisms groups.
We find sharp estimates for the growth sequenc...
We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in C^n is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behaviour of Oh's spectral sequence with respect to this product. As further applications we pro...
In this paper we find new topological restrictions on Lagrangian submanifolds of cotangent bundles of spheres and of Lens spaces. Comment: 6 pages
Citations
... It is clear that SG ω (Σ g ) ⊆ F SHomeo(Σ g ) but to our knowledge it is not known whether the groups SG ω (Σ g ) and F SHomeo(Σ g ) are distinct groups. In [5], using a sequence of spectral invariants the author showed that Hameo(S 2 , ω) is a proper normal subgroup F Homeo(S 2 ). ...
... A powerful tool for proving flexibility statements in C 0 symplectic geometry is the quantitative h-principle introduced first for symplectic 2-discs in [BO16], and more recently for subcritical isotropic discs in [BO21]. However, contact quantitative h-principle has not been established previously. ...
... 7, by combining the machinery we have developed, we give a purely sheaf-theoretic proof of the following Arnold-type theorem for non-smooth objects (cf. Buhovsky-Humilière-Seyfaddini [11]). We let 0 M denote the zero-section of T * M. ...
... A C 0 -almost periodic Hamiltonian diffeomorphism or C 0 -a.i. is automatically γ-almost periodic or, respectively, γ-a.i. when M is symplectically aspherical, [6], and also for some other classes of monotone symplectic manifolds M including CP n , [41]. However, we are not aware of any example of Hamiltonian C 0 -a.i.'s on a symplectically aspherical manifold and hypothetically such maps do not exist. ...
Reference: On the growth of the Floer barcode
... , l. Similar results have been obtained in [14][15][16]. ...
... The Poisson bracket invariant is known to be strictly positive when the cover consists of displaceable sets. Polterovich conjectured a lower bound for this invariant, which can be interpreted as an uncertainty principle: This conjecture was proved for the case where M is any surface in [BLT20], and for surfaces other than the sphere in [Pay18]. In higher dimensions the conjecture is still open. ...
... A few results are deterministic and so their proofs require no modification. These facts require notions of boundary and interior for tilted rectangles; the notion of the western boundary, at least implicitly, was used in [Buh+22], where the question of unique continuation was studied under the assumption V ≡ 0. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272474319355915-1441974348457_Q64/Eugenia-Malinnikova.jpg)
... Theorem 3. Let (M, ω) be a negatively monotone symplectic manifold and suppose U i ⊂ M are disjoint domains with incompressible contact type boundaries, such that the contact Conley-Zehnder index of every Reeb orbit is non-negative. Then, for any collection of 2 We remark that one can perturb the Liouville vector field to make the Reeb flow non-denegerate, see Section 2. ...
... In Sect. 6 we conclude with a discussion of the existence and covariance structure of random potentials, that is, solutions to the equation = 2π(n − c m). ...
... Conversely, Entov-Polterovich-Py point out that on Ham.D 2n .1//, the group of compactly supported Hamiltonian diffeomorphisms of the closed unit ball D 2n .1/ in R 2n , the Hofer metric is not C 0 -continuous. For some striking results that demonstrate rigidity and flexibility of symplectic objects with respect to C 0 -topology, see [13,14] and [36]. ...
