Floer homology in disc bundles and symplectically twisted geodesic flows
Abstract
We show that if K: P R is an autonomous Hamiltonian on a symplectic manifold (P,) 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 G\"urel when |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.