Non-uniqueness of geodesic limits and a question of Grayson and Gage

2026/08/05 by Shrey Aryan, Tang-Kai Lee
Mathematics · #math.DG #math.AP

paper · pdf

32 Pages

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on \mathbbS² and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

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