Multiplicity and stability of closed geodesics on positively curved Finsler $4$-spheres

TL;DR

This paper investigates the number and stability of closed geodesics on positively curved Finsler 4-spheres, establishing conditions under which multiple geodesics exist and describing their stability types.

Contribution

It provides new existence and stability results for closed geodesics on Finsler 4-spheres under specific curvature and reversibility conditions.

Findings

At least four prime closed geodesics exist under certain curvature conditions.

If exactly three exist, two are irrationally elliptic and non-hyperbolic.

Conditions on reversibility and flag curvature are crucial for these results.

Abstract

In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ with $\lambda<\frac{3}{2}$, either there exist at least four prime closed geodesics, or there exist exactly three prime non-hyperbolic closed geodesics and at least two of them are irrationally elliptic.