Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds

Abstract

We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,\omega )$ and an open subset $N\subset M$ endowed with a Hamiltonian free circle action $\varphi$ then $N$ has bounded Hofer-Zehnder $G_{\varphi }$-semicapacity, where $G_{\varphi }\subset {\pi }_1(N)$ is the subgroup generated by the homotopy class of the orbits of $\varphi$. In particular, $N$ has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle $T^{\ast }G$ of any compact Lie group $G$ has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-Gürel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.

arXiv comment19 pages, 4 figures. Revised version

References in corpus (1)

• A refinement of the Hofer-Zehnder theorem on the existence of closed trajectories near a hypersurface

Cited by in corpus (7)

• The contact property for nowhere vanishing magnetic fields on the two-sphere
• Applications of Hofer's geometry to Hamiltonian dynamics
• Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles
• Hofer-Zehnder capacity and Hamiltonian circle actions
• Hofer-Zehnder capacity of standard cotangent bundles
• Floer homology in disc bundles and symplectically twisted geodesic flows
• Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic submanifolds