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Symplectic topology of Mañé's critical values

Kai Cieliebak, Urs Frauenfelder and Gabriel P Paternain

Geometry & Topology 14 (2010) 1765–1870

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mañé critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and that it is invariant under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k > c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is nonzero for energy levels k > c but vanishes for k < c, so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 14–pinched negative curvature and nonexact magnetic fields all sufficiently high energy levels are nonstable, provided that the dimension of the base manifold is even and different from two.

Mañé' critical value, magnetic field, Rabinowitz Floer homology, stable Hamiltonian structure
Mathematical Subject Classification 2000
Primary: 53D40
Secondary: 37D40
Received: 3 November 2009
Revised: 25 April 2010
Accepted: 26 May 2010
Published: 21 July 2010
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Yasha Eliashberg
Kai Cieliebak
Mathematisches Institut
Ludwig-Maximilians-Universität München
Theresienstr 39
80333 München
Urs Frauenfelder
Department of Mathematics and Research Institute of Mathematics
Seoul National University
San56-1 Shinrim-dong Kwanak-gu
Seoul 151-747
Gabriel P Paternain
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
United Kingdom