Two elliptic closed geodesics on positively curved Finsler spheres

TL;DR

This paper proves that on certain positively curved Finsler spheres, either infinitely many closed geodesics exist or there are at least two elliptic closed geodesics with specific eigenvalue properties, advancing understanding of geodesic multiplicity.

Contribution

It establishes a new dichotomy for closed geodesics on Finsler spheres under curvature conditions, highlighting the existence of elliptic geodesics with irrational eigenvalues.

Findings

Either infinitely many closed geodesics exist or at least two elliptic ones with irrational eigenvalues.

Provides conditions on reversibility and flag curvature for geodesic multiplicity.

Advances the understanding of geodesic structure on positively curved Finsler spheres.

Abstract

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^{n},F)$ with reversibility $\lm$ and flag curvature $K$ satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincar\'{e} map has at least one eigenvalue of the form $e^{\sqrt{-1}\th}$ with $\th$ being an irrational multiple of $\pi$.