Floer homology in disc bundles and symplectically twisted geodesic flows

2008/07/10 by Michael Usher · 1 citation
Mathematics · #math.SG #math.DG #math.DS

paper · pdf

published as J. Mod. Dyn. 3 (2009), no. 1, 61--101 · 37 pages, one figure

arxiv created 2008/07/10 · arxiv updated 2011/01/27

Abstract

We show that if K: P → R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c₁(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 Gürel when Ω|_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.

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