A homological criterion for the almost-existence of Hamiltonian chords

TL;DR

This paper introduces a homological criterion based on wrapped Floer homology that guarantees the near-existence of Hamiltonian chords in Liouville domains, linking capacity notions with Floer homology vanishing.

Contribution

It develops a new homological criterion involving a relative coisotropic Hofer-Zehnder capacity to determine the existence of Hamiltonian chords.

Findings

Wrapped Floer homology vanishing implies Hamiltonian chords exist near any hypersurface.

Introduces a relative coisotropic Hofer-Zehnder capacity and compares it with other capacities.

Provides a criterion connecting Floer homology and Hamiltonian dynamics.

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.