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A homological criterion for the almost-existence of Hamiltonian chords

Antoine Rodrigues

TL;DR

The paper develops a homological criterion for the almost-existence of Hamiltonian chords near a given energy level inside a Liouville domain. Central to the approach is a relative Lagrangian Hofer–Zehnder capacity derived from wrapped Floer homology, together with spectral invariants that bound this capacity by a wrapped-Floer–based symplectic capacity $c_{SH}$. The main result shows that if there exists a Lagrangian homology class $eta eq0$ with $i_L(eta)=0$ but whose shriek image remains nontrivial in a relative setting, then almost all nearby energy levels carry a Hamiltonian chord; a key corollary is that if $SH_*(W,L)=0$, almost-existence holds for any hypersurface transverse to $L$. The framework unifies capacities and homologies to produce broad almost-existence results, with concrete implications for subcritical/flexible Weinstein domains and displaceable Lagrangians.

Abstract

We establish a criterion on wrapped Floer homology of an exact Lagrangian sub- manifold in a Liouville domain, which ensures the almost-existence of Hamiltonian chords near a given energy level. To this purpose we introduce a relative version of the coisotropic Hofer-Zehnder capacity, and compare it with some other capacities that are defined using filtered wrapped Floer homology. Our main application is the almost- existence of Hamiltonian chords near any hypersurface if the wrapped Floer homology vanishes.

A homological criterion for the almost-existence of Hamiltonian chords

TL;DR

The paper develops a homological criterion for the almost-existence of Hamiltonian chords near a given energy level inside a Liouville domain. Central to the approach is a relative Lagrangian Hofer–Zehnder capacity derived from wrapped Floer homology, together with spectral invariants that bound this capacity by a wrapped-Floer–based symplectic capacity . The main result shows that if there exists a Lagrangian homology class with but whose shriek image remains nontrivial in a relative setting, then almost all nearby energy levels carry a Hamiltonian chord; a key corollary is that if , almost-existence holds for any hypersurface transverse to . The framework unifies capacities and homologies to produce broad almost-existence results, with concrete implications for subcritical/flexible Weinstein domains and displaceable Lagrangians.

Abstract

We establish a criterion on wrapped Floer homology of an exact Lagrangian sub- manifold in a Liouville domain, which ensures the almost-existence of Hamiltonian chords near a given energy level. To this purpose we introduce a relative version of the coisotropic Hofer-Zehnder capacity, and compare it with some other capacities that are defined using filtered wrapped Floer homology. Our main application is the almost- existence of Hamiltonian chords near any hypersurface if the wrapped Floer homology vanishes.

Paper Structure

This paper contains 16 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Structure of the paper
  5. A relative version of the Lagrangian Hofer-Zehnder capacity
  6. Conventions and overview of wrapped Floer homology
  7. General setting
  8. Action functional
  9. Spaces of trajectories
  10. Floer homology
  11. Wrapped Floer homology
  12. From symplectic homology to Morse homology
  13. The homological criterion
  14. A spectral invariant
  15. The spectral capacity
  16. ...and 1 more sections

Key Result

Theorem 1.2

dg-floer2 Let $\Sigma$ be a hypersurface bounding a relatively compact open subset $U$. Suppose there exists a homology class $\alpha \in H_*(W,\partial W)$ such that $i_W(\alpha)=0$ and $j_{U!}(\alpha)\neq 0$. Then the almost-existence property for contractible periodic orbits holds in a neighborho

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 19 more