Besicovitch-type inequality for closed geodesics on 2-dimensional spheres

TL;DR

This paper establishes a universal inequality relating the lengths of two distinct closed geodesics on any Riemannian 2-sphere, linking their product to the sphere's area, extending geometric inequalities in differential geometry.

Contribution

It proves a Besicovitch-type inequality for closed geodesics on 2-spheres, providing a new universal bound connecting geodesic lengths and surface area.

Findings

Existence of two closed geodesics with bounded length product

Universal constant C independent of the metric

Relation between geodesic lengths and sphere area

Abstract

We prove the existence of a constant $C > 0$ such that for any Riemannian metric $g$ on a 2-dimensional sphere $S^2$, there exist two distinct closed geodesics with lengths $L_{1}$ and $L_{2}$ satisfying $L_{1} L_{2} \leq C \cdot \operatorname{Area}(S^2, g)$.