On the rigidity of Zoll magnetic systems on surfaces

Abstract

Abstract In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level below the Mañe critical value. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.