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

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.

37 pages. arXiv admin note: text overlap with arXiv:1508.05577; text overlap with arXiv:1712.07000, arXiv:1804.06193, arXiv:1804.08452, arXiv:1803.08350 by other authors