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Multiple closed geodesics on Finsler $3$-dimensional sphere

Huagui Duan, Zihao Qi

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.

Multiple closed geodesics on Finsler $3$-dimensional sphere

Abstract

In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric on 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 . In this paper, we proved this conjecture for bumpy Finsler if the Morse index of any prime closed geodesic is nonzero.
Paper Structure (4 sections, 46 equations)

This paper contains 4 sections, 46 equations.

Table of Contents

  1. Introduction and main result
  2. Critical modules of closed geodesics and Morse Theory
  3. The common index jump theorem for symplectic paths
  4. Proof of Theorem 1.1