Floer homology in disc bundles and symplectically twisted geodesic flows

Details Floer homology in disc bundles and symplectically twisted geodesic flows Floer homology in disc bundles and symplectically twisted geodesic flows

2008-07-10

arxiv.org/abs/0807.1694v1

J. Mod. Dyn. 3 (2009), no. 1, 61--101

Mathematics

Symplectic Geometry

37 pages, one figure

Scientific paper

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.

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