On the rigidity of Zoll magnetic systems on surfaces

TL;DR

This paper investigates the rigidity of Zoll magnetic systems on closed surfaces, characterizing those with constant curvature and magnetic functions as uniquely Zoll systems, and studies the persistence of closed geodesics under perturbations.

Contribution

It provides a characterization of Zoll magnetic systems on surfaces of positive genus and proves the stability of closed geodesics under magnetic perturbations.

Findings

Magnetic systems with constant curvature and magnetic functions are uniquely Zoll.

Closed geodesics persist under magnetic perturbations.

Characterization of Zoll magnetic systems on surfaces of positive genus.

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. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.