JournalscmhVol. 84, No. 4pp. 865–907

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

  • Viktor L. Ginzburg

    UC Santa Cruz, USA
  • Başak Z. Gürel

    Vanderbilt University, Nashville, USA
Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem cover
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Abstract

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.

Cite this article

Viktor L. Ginzburg, Başak Z. Gürel, Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem. Comment. Math. Helv. 84 (2009), no. 4, pp. 865–907

DOI 10.4171/CMH/184