Commentarii Mathematici Helvetici
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Published online: 2009-12-23
Periodic orbits of twisted geodesic flows and the Weinstein–Moser theoremViktor L. Ginzburg and Başak Z. Gürel (1) UC Santa Cruz, USA
(2) Vanderbilt University, Nashville, USA
In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse–Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein–Moser theorem. The proof of the generalized Weinstein–Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.
Keywords: Twisted geodesic flows, periodic orbits, Floer homology, Sturm theory
Ginzburg Viktor, Gürel Başak: Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem. Comment. Math. Helv. 84 (2009), 865-907. doi: 10.4171/CMH/184