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Fabian Ziltener

Publications and source records attributed to Fabian Ziltener.

At least 19 recordsLinked to original sources
arXiv2024

Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets

We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$ equals $π$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.

math.SG↗
arXiv2023

Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture

Let $G$ be a compact and connected Lie group. The Hamiltonian $G$-model functor maps the category of symplectic representations of closed subgroups of $G$ to the category of exact Hamiltonian $G$-actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian $G$-actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian $G$-actions on contractible manifolds are exactly the symplectic $G$-representations, up to isomorphism.

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arXiv2022

Recognition of objects through symplectic capacities

We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form $(M, ω)$, such that $M$ is a compact and 1-connected manifold, $ω$ is an exact symplectic form on $M$, and there exists a boundary component of $M$ with negative helicity. The set of generalized symplectic capacities is thus a complete invariant for such categories. This answers a question by Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for recognition results for manifolds of dimension 2, ellipsoids, and polydiscs in $\mathbb{R}^4$. Strikingly, our result holds more generally for differential form categories. Recognition of objects is therefore not a symplectic phenomenon. We also prove a version of the result for normalized capacities.

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arXiv2021

A relative Hofer estimate and the asymptotic Hofer-Lipschitz constant

Let $(M,ω)$ be a symplectic manifold and $U\subseteq M$ an open subset. We study the natural inclusion of the compactly supported Hamiltonian group of $U$ in the compactly supported Hamiltonian group of $M$. The main result is an upper bound for this map in terms of the Hofer norms for $U$ and $M$. Applications are upper bounds on the asymptotic Hofer-Lipschitz constant and the relative Hofer diameter of $U$. The first bound is often sharp and the second one is often sharp up to a factor of 2.

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arXiv2021

Generating sets and representability for symplectic capacities

K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for two results of D. McDuff about the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating set of capacities on a given symplectic category is uncountable, provided that the category contains a one-parameter family of symplectic manifolds that is ``strictly volume-increasing'' and ``embedding-capacity-wise constant''. It follows that the Ekeland-Hofer capacities and the volume capacity do not finitely differentiably generate all generalized capacities on the category of ellipsoids. This answers a variant of a question of CHLS. In addition, we prove that if a given symplectic category contains a certain one-parameter family of objects, then almost no normalized capacity is domain- or target-representable. This provides some solutions to two central problems of CHLS.

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arXiv2020

Note on coisotropic Floer homology and leafwise fixed points

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $ϕ=(ϕ^t)$ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $(N,ϕ)$-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold. Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine. The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

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arXiv2017

Leafwise fixed points for $C^0$-small Hamiltonian flows

Consider a closed coisotropic submanifold $N$ of a symplectic manifold $(M,ω)$ and a Hamiltonian diffeomorphism $ϕ$ on $M$. The main result of this article states that $ϕ$ has at least the cup-length of $N$ many leafwise fixed points w.r.t. $N$, provided that it is the time-1-map of a global Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. If $(ϕ,N)$ is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of $N$. The nondegeneracy condition is generically satisfied. This appears to be the first leafwise fixed point result in which neither $ϕ\big|_N$ is assumed to be $C^1$-close to the inclusion $N\to M$, nor $N$ to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the $C^0$-condition on $ϕ$ cannot be replaced by the assumption that $ϕ$ is Hofer-small.

math.SG↗
arXiv2015

On the strict Arnold chord property and coisotropic submanifolds of complex projective space

Let $α$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for $α$ at least twice if all characteristics are closed and of the same period, and $α$ embeds nicely into the product of $\mathbb{R}^{2n}$ and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most $π/2$. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.

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arXiv2013

Discontinuous symplectic capacities

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.

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arXiv2012

Morphisms of CohFT algebras and quantization of the Kirwan map

We introduce a notion of morphism of CohFT algebras, based on the analogy with A-infinity morphisms. We discuss a "quantization" of the classical Kirwan morphism to a morphism of CohFT algebras from the equivariant quantum cohomology of a G-variety to the quantum cohomology of its git or symplectic quotient, and an example relating to the orbifold quantum cohomology of a compact toric orbifold. Finally we identify the space of Cartier divisors on the moduli space of scaled marked curves; these appear in the splitting axiom.

math.AG↗
arXiv2012

A Quantum Kirwan Map, I: Fredholm Theory

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,ω)$. Under some assumptions on $(M,ω)$ and the action, D. A. Salamon conjectured that counting gauge equivalence classes of symplectic vortices on the plane $R^2$ gives rise to a quantum deformation $Qκ_G$ of the Kirwan map. This article is the first of three, whose goal is to define $Qκ_G$ rigorously. Its main result is that the vertical differential of the vortex equations over $R^2$ (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially, the map $Qκ_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization (see [Woodward-Ziltener]).

math.SG↗
arXiv2012

A Quantum Kirwan Map, II: Bubbling

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,ω)$. Under suitable assumptions, counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$ conjecturally gives rise to a quantum deformation $Qk_G$ of the Kirwan map. This is the second of a series of articles, whose goal is to define $Qk_G$ rigorously. The main result is that every sequence of vortices with uniformly bounded energies has a subsequence that converges to a genus 0 stable map of vortices on $R^2$ and holomorphic spheres in the symplectic quotient. Potentially, the map $Qk_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization.

math.SG↗
arXiv2012

A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane

Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M,ω)$. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of $(M,ω)$ to the Gromov-Witten theory of the symplectic quotient. The morphism should be a deformation of the Kirwan map. The idea, due to D. A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane $C$. The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case. The first one states that every sequence of equivalence classes of vortices over the plane has a subsequence that converges to a new type of genus zero stable map, provided that the energies of the vortices are uniformly bounded. Such a stable map consists of equivalence classes of vortices over the plane and holomorphic spheres in the symplectic quotient. The second main result is that the vertical differential of the vortex equations over the plane (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially the quantum Kirwan map can be used to compute the quantum cohomology of symplectic quotients.

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arXiv2012

Coisotropic Displacement and Small Subsets of a Symplectic Manifold

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal symplectic area of a regular coisotropic submanifold of dimension $d$.

math.DG↗
arXiv2012

A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$-th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.

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