Multiple closed geodesics on Finsler $3$-dimensional sphere

TL;DR

This paper proves that on bumpy Finsler 3-spheres with prime closed geodesics having nonzero Morse index, the minimal number of such geodesics is four, confirming a conjecture by Anosov.

Contribution

It confirms Anosov's conjecture for a class of bumpy Finsler 3-spheres with nonzero Morse index for prime closed geodesics.

Findings

Minimum of four prime closed geodesics on the specified Finsler 3-spheres.

Validation of Anosov's conjecture in this setting.

Extension of previous results on closed geodesics.

Abstract

In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric on $S^3$ with exactly four prime closed geodesics. And then Anosov conjectured that four should be the optimal lower bound of the number of prime closed geodesics on every Finsler $S^3$. In this paper, we proved this conjecture for bumpy Finsler $S^{3}$ if the Morse index of any prime closed geodesic is nonzero.