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1 November 2021 Uniqueness of asymptotically conical tangent flows
Otis Chodosh, Felix Schulze
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Duke Math. J. 170(16): 3601-3657 (1 November 2021). DOI: 10.1215/00127094-2020-0098

Abstract

Singularities of the mean curvature flow of an embedded surface in R3 are expected to be modeled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity.

In all dimensions, assuming that the singularity is of multiplicity 1, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding and Minicozzi. We show here the uniqueness of multiplicity-1 asymptotically conical tangent flows for mean curvature flow of hypersurfaces.

In particular, this implies that when a mean curvature flow has a multiplicity-1 conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to “flow through” such a singularity.

Citation

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Otis Chodosh. Felix Schulze. "Uniqueness of asymptotically conical tangent flows." Duke Math. J. 170 (16) 3601 - 3657, 1 November 2021. https://doi.org/10.1215/00127094-2020-0098

Information

Received: 14 June 2019; Revised: 2 December 2020; Published: 1 November 2021
First available in Project Euclid: 14 October 2021

Digital Object Identifier: 10.1215/00127094-2020-0098

Subjects:
Primary: 53C44
Secondary: 35B35

Keywords: asymptotically conical shrinkers , Mean curvature flow , uniqueness of tangent flows

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 16 • 1 November 2021
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