On a systolic inequality for closed magnetic geodesics on surfaces

TL;DR

This paper establishes bounds on the magnetic length of closed geodesics on surfaces using a systolic inequality, applicable when the curvature is near Zoll or sufficiently large.

Contribution

It introduces a novel application of systolic inequalities to magnetic geodesics on surfaces, extending previous geometric bounds.

Findings

Bounds on magnetic length for prescribed curvature near Zoll cases

Bounds on magnetic length for large prescribed curvature

Application of contact and odd-symplectic form inequalities

Abstract

We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close to a Zoll one or large enough.