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arXiv · 2608.04855

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

Shrey Aryan·Tang-Kai Lee
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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 $\mathbb{S}^2$ 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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Shrey Aryan, Tang-Kai Lee. 2026-08-05. Non-uniqueness of geodesic limits and a question of Grayson and Gage. https://arxiv.org/abs/2608.04855

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