Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds
Hofer-Zehnder capacity | periodic orbits | Hamiltonian circle actions | Weinstein conjecture
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 $T^*G$ 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-Gúrel: almost all levels of a function defined on a neighborhood of a closed symplectic submanifold $S$ in a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function is constant on $S$.