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

In this paper, we proved that for every Finsler metric on Sⁿ (n 4) with reversibility and flag curvature K satisfying (2n-3n-1) ² (+1) ²<K 1 and <n-1n-2, there exist at least n prime closed geodesics on (Sⁿ, 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 Sⁿ is finite, then there exist at least 22-1 non-hyperbolic closed geodesics.