Closed mean curvature flows with asymptotically conical singularities
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.
Cite
@article{arxiv.2405.15577,
title = {Closed mean curvature flows with asymptotically conical singularities},
author = {Tang-Kai Lee and Xinrui Zhao},
journal= {arXiv preprint arXiv:2405.15577},
year = {2024}
}
Comments
23 pages; v2: A new result was added as a corollary thanks to Alec Payne's suggestion; v3: slight modification with references updated