Integrable magnetic flows on the two-torus: Zoll examples and systolic inequalities

TL;DR

This paper explores integrable magnetic flows on the two-torus, providing new Zoll examples where all magnetic geodesics are closed and establishing a sharp systolic inequality for systems with a global surface of section.

Contribution

It constructs the first non-trivial Zoll examples of magnetic flows on the two-torus and proves a new systolic inequality for integrable magnetic systems with a global surface of section.

Findings

Existence of non-trivial Zoll magnetic flows on the two-torus

A sharp systolic inequality for integrable magnetic systems with a global surface of section

Characterization of magnetic geodesics with all closed orbits

Abstract

In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.