Closed geodesics on positively curved Finsler spheres

Abstract

In this paper, we prove that for every Finsler $n$-sphere $(S^n,F)$ for $n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda }{\lambda +1}{)}^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form $\exp (\pi i\mu )$ with an irrational $\mu$. Furthermore, there always exist three prime closed geodesics on any $(S^3,F)$ satisfying the above pinching condition.

41 pages. Revised version. To appear in Adv. Math