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

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.