Searcharxiv⌕ Search

arXiv subjects

Felix Schlenk

Publications and source records attributed to Felix Schlenk.

At least 19 recordsLinked to original sources
arXiv2025

Markov staircases

Rational homology ellipsoids are certain Liouville domains diffeomorphic to rational homology balls and having Lagrangian pin-wheels as their skeleta. From the point of view of almost toric fibrations, they are a natural generalisation of usual symplectic ellipsoids. We study symplectic embeddings of rational homology ellipsoids into the complex projective plane and we show that for each Markov triple, this problem gives rise to an infinite staircase. A key ingredient in the proof is the result that any two such embeddings are Hamiltonian isotopic. We also prove constraints on sizes for pairs of disjoint embeddings.

math.SG↗
arXiv2024

Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$

The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.

math.SG↗
arXiv2024

Lagrangian knots and unknots -- an essay

In this essay dedicated to Yakov Eliashberg we survey the current state of the field of Lagrangian (un)knots, reviewing some constructions and obstructions along with a number of unsolved questions. The appendix by Georgios Dimitroglou Rizell provides a new take on local Lagrangian knots.

math.SG↗
arXiv2023

Entropy collapse versus entropy rigidity for Reeb and Finsler flows

On every closed contact manifold there exist contact forms with volume one whose Reeb flows have arbitrarily small topological entropy. In contrast, for many closed manifolds there is a uniform positive lower bound for the topological entropy of (not necessarily reversible) normalized Finsler geodesic flows.

math.DS↗
arXiv2022

Pinwheels as Lagrangian barriers

The complex projective plane CP^2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of CP^2, meaning that pinwheels are Lagrangian barriers in the sense of Biran. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below 3.

math.SG↗
arXiv2020

Hamiltonian delay equations -- examples and a lower bound for the number of periodic solutions

We describe a variational approach to a notion of Hamiltonian delay equations. Our delay Hamiltonians are of product form. We consider several examples. For closed symplectically aspherical symplectic manifolds $(M,ω)$ we prove that for generic delay Hamiltonians the number of 1-periodic solutions of the Hamiltonian delay equation is at least the sum of the Betti numbers of $M$, extending the proof of the Arnold conjecture to the case with delay.

math.DS↗
arXiv2018

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.

math.DS↗
arXiv2017

Floer homologies, with applications

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists, and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry.

math.SG↗
arXiv2016

$S^1$-equivariant Rabinowitz-Floer homology

We define the $S^1$-equivariant Rabinowitz-Floer homology of a bounding contact hypersurface $Σ$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $Σ$ is displaceable. In the appendix we describe an approach to transversality for Floer homologies for which the moduli space $\widehat M_J$ of all gradient flow lines is compact for some almost complex structure $J$. This approach uses a large set of perturbations, namely vector fields on the loop space, and selects from the possibly non-compact perturbed moduli spaces a part near $\widehat M_J$ that turns out to be compact for small enough perturbations.

math.SG↗
arXiv2016

Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

In previous work, the second author and Müller determined the function $c(a)$ giving the smallest dilate of the polydisc $P(1,1)$ into which the ellipsoid $E(1,a)$ symplectically embeds. We determine the function of two variables $c_b(a)$ giving the smallest dilate of the polydisc $P(1,b)$ into which the ellipsoid $E(1,a)$ symplectically embeds for all integers $b \geqslant 2$. It is known that for fixed $b$, if $a$ is sufficiently large then all obstructions to the embedding problem vanish except for the volume obstruction. We find that there is another kind of change of structure that appears as one instead increases $b$: the number-theoretic "infinite Pell stairs" from the $b=1$ case almost completely disappears (only two steps remain), but in an appropriately rescaled limit, the function $c_b(a)$ converges as $b$ tends to infinity to a completely regular infinite staircase with steps all of the same height and width.

math.SG↗
arXiv2015

Lagrangian product tori in tame symplectic manifolds

Product Lagrangian tori in standard symplectic space $R^{2n}$ were classified up to symplectomorphism in [Che96]. We extend this classification to tame symplectically aspherical symplectic manifolds. We show by examples that the asphericity assumption cannot be omitted.

math.SG↗
arXiv2014

Shortest closed billiard orbits on convex tables

Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular $n$-gon $R_n$ is 2-bounce for $n \ge 4$, with length twice the width of $R_n$. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domain $T \times B^2$ in the standard symplectic vector space $\mathbb{R}^4$. Our method is based on the work of Bezdek-Bezdek and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.

math.DG↗
arXiv2013

Healthy vector spaces and spicy Hopf algebras (with applications to the growth rate of geodesic chords and to intermediate volume growth on manifolds of non-finite type)

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geodesic flows, these lower bounds are: (i) For any Riemannian metric on M, any pair of non-conjugate points p,q in M, and every component C of the space of paths from p to q, the number of geodesics in C of length at most T grows at least like e^{\sqrt T}. (ii) The exponent of the volume growth of any geodesic flow on M is at least 1/2. We obtain these results by combining new algebraic results on the growth of certain filtered Hopf algebras with known results on Floer homology.

math.AT↗
arXiv2013

Slow volume growth for Reeb flows on spherizations and contact Bott--Samelson theorems

We give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S*M,ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our uniform bound is in terms of the polynomial growth of the homology of the based loops space of M. As an application, we extend the Bott--Samelson theorem from geodesic flows to Reeb flows: If (S*M,ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fibre S*_q M, then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M is the one of a compact rank one symmetric space.

math.DS↗
arXiv2013

The Gromov width of 4-dimensional tori

We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of the 4-torus with its given linear symplectic form.

math.SG↗