Index, Intersections, and Multiplicity of Min-Max Geodesics

TL;DR

This paper establishes bounds on the Morse index and intersections of min-max geodesics on surfaces, and shows that multiplicity one is not generic by constructing specific metric examples.

Contribution

It provides new bounds for geodesic properties and demonstrates non-generic multiplicity phenomena through explicit metric constructions.

Findings

Upper bounds for Morse index of min-max geodesics

Bounds on the number of intersections of geodesics

Existence of metrics with multiple copies of a single geodesic

Abstract

We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on $S^2$ for which the $p$-width is only achieved by $p$ copies of a single geodesic.