Hofer-Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces

TL;DR

This paper calculates the Hofer-Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces using Hamiltonian circle actions and pseudo-holomorphic curve theory, providing bounds based on flow periodicity and Gromov-Witten invariants.

Contribution

It introduces a method to compute Hofer-Zehnder capacity for magnetic tangent bundles on constant curvature surfaces by leveraging magnetic geodesic flow properties and pseudo-holomorphic curve techniques.

Findings

Lower bound from Hamiltonian oscillation

Upper bound from Lu's capacity bounds

Use of Gromov-Witten invariants for non-vanishing results

Abstract

We compute the Hofer-Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer-Zehnder capacity. The upper bound is obtained from Lu's bounds of the Hofer-Zehnder capacity using the theory of pseudo-holomorphic curves. In our case the gradient spheres of the Hamiltonian will give rise to the non-vanishing Gromov-Witten invariant needed.