Symplectic topology of Ma\~n\'e's critical values

TL;DR

This paper investigates the symplectic topology and dynamics of energy hypersurfaces in twisted cotangent bundles, focusing on the behavior at Mañé's critical value using Rabinowitz Floer homology, revealing non-displaceability and stability properties.

Contribution

It introduces Rabinowitz Floer homology for certain hypersurfaces and demonstrates its invariance, providing new insights into the topology and dynamics at Mañé's critical value in twisted cotangent bundles.

Findings

Rabinowitz Floer homology is defined for stable tame or virtually contact hypersurfaces.

Energy levels above Mañé's critical value are non-displaceable when the configuration space has negative curvature.

High energy levels are non-stable under certain curvature and magnetic field conditions, especially in even-dimensional base manifolds.

Abstract

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 Mane 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 it is invariant under 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 non-zero 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 1/4-pinched negative curvature and non-exact magnetic fields all sufficiently high energy levels are non-stable, provided that the dimension of the base manifold is even and different from two.