Index, Intersections, and Multiplicity of Min-Max Geodesics

2024/10/03 by Jared Marx-Kuo, Marx-Kuo, Jared, Lorenzo Sarnataro +3
Engineering · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.02580

openalex publication_date 2024/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 S² for which the p-width is only achieved by p copies of a single geodesic.

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