Floer homology in disc bundles and symplectically twisted geodesic flows

TL;DR

This paper proves the existence of contractible periodic orbits in certain Hamiltonian flows on symplectic manifolds, extending previous results by constructing a specialized Floer homology in the symplectic normal disc bundle.

Contribution

It introduces a new approach to analyze Floer homology in symplectic normal disc bundles, generalizing prior work to broader conditions.

Findings

Existence of contractible periodic orbits on small energy levels.

Extension of Ginzburg and G"urel's results to more general settings.

Construction of filtered Floer homology in symplectic normal disc bundles.

Abstract

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on the cotangent bundle of M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and G\"urel when \Omega|_M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disc bundle to M.