Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits

TL;DR

This paper proves that in certain symplectic manifolds with a noncontractible circle action, the Hofer-Zehnder capacity of an open set is bounded by the Hofer norm of the generating Hamiltonian, using action selectors.

Contribution

It establishes a capacity bound in aspherical symplectic manifolds with noncontractible orbits, extending the understanding of symplectic capacities and Hamiltonian dynamics.

Findings

Hofer-Zehnder capacity is bounded by the Hofer norm of the Hamiltonian.

Uses a variant of the energy-capacity inequality with action selectors.

Applicable to aspherical symplectic manifolds with noncontractible circle actions.

Abstract

Let $(M,\omega)$ be an aspherical symplectic manifold, which is closed or convex. Let $U$ be an open set in $M$, which admits a circle action generated by an autonomous Hamiltonian $H \in C^\infty(U)$, such that each orbit of the circle action is not contractible in $M$. Under these assumptions, we prove that the Hofer-Zehnder capacity of $U$ is bounded by the Hofer norm of $H$. The proof uses a variant of the energy-capacity inequality, which is proved by the theory of action selectors.