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arXiv · 2405.15577

Closed mean curvature flows with asymptotically conical singularities

Tang-Kai Lee·Xinrui Zhao
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Abstract

In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Wa\.zewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent--Ilmanen--Vel\'azquez and Chodosh--Daniels-Holgate--Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans--Spruck.

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Tang-Kai Lee, Xinrui Zhao. 2024-05-24. Closed mean curvature flows with asymptotically conical singularities. https://arxiv.org/abs/2405.15577

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