Bumpy metrics on spheres and minimal index growth

TL;DR

This paper simplifies the proof of the existence of multiple closed geodesics on spheres with bumpy Finsler metrics and explores metrics with finitely many geodesics, including constructions with prescribed properties.

Contribution

It provides a simplified proof of the existence of multiple closed geodesics on spheres with bumpy Finsler metrics and discusses perturbations of Katok metrics with finitely many geodesics.

Findings

Existence of two geometrically distinct closed geodesics on spheres with bumpy Finsler metrics.

A covering of minimal index growth leads to a contradiction for infinite index bounds.

Construction of metrics on S^2 with only two non-intersecting closed geodesics of arbitrary large length.

Abstract

The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed geodesics of index $\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\left(c^{rm}\right)\le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky--Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large $L>0$ we obtain on $S^2$ a metric of positive flag curvature carrying only two closed geodesics of length $<L$ which do not intersect.