Lengths of three simple periodic geodesics on a Riemannian $2$-sphere

TL;DR

This paper establishes upper bounds on the lengths of three simple periodic geodesics on a Riemannian 2-sphere, relating them to the sphere's diameter and area, and improves bounds for thin spheres.

Contribution

It provides explicit length bounds for three simple periodic geodesics on a Riemannian 2-sphere, refining classical results and introducing bounds depending on area and diameter.

Findings

Existence of three simple periodic geodesics with lengths ≤ 20 times the diameter.

Upper bounds on geodesic lengths depending on area and diameter.

Asymptotically optimal bounds for thin spheres when area is much less than diameter squared.

Abstract

Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for "thin" spheres, when the area $A$ is much less than $d^2$, where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when ${A\over d^2}\longrightarrow 0$.