On the minimal number of closed geodesics on positively curved Finsler spheres

TL;DR

This paper proves the existence of at least n prime closed geodesics on certain positively curved Finsler spheres in dimensions n≥4, confirming a conjecture for even n and providing bounds on non-hyperbolic geodesics.

Contribution

It establishes new lower bounds on the number of closed geodesics on positively curved Finsler spheres under specific curvature and reversibility conditions, solving a longstanding conjecture for even dimensions.

Findings

At least n prime closed geodesics exist under given conditions.

Finiteness of closed geodesics implies at least 2[n/2]-1 non-hyperbolic geodesics.

Confirmed a conjecture of Katok and Anosov for even-dimensional spheres.

Abstract

In this paper, we proved that for every Finsler metric on $S^n$ $(n\ge 4)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{2n-3}{n-1})^2 (\frac{\lambda}{\lambda+1})^2<K\le 1$ and $ \lambda<\frac{n-1}{n-2} $, there exist at least $n$ prime closed geodesics on $(S^n,F)$, which solved a conjecture of Katok and Anosov for such positivley curved spheres when $n$ is even. Furthermore, if the number of closed geodesics on such positively curved Finsler $S^n$ is finite, then there exist at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic closed geodesics.