Uniqueness of asymptotically conical tangent flows

Abstract

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled 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 the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one 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.

40 pages. Final version, to appear in Duke Mathematical Journal

References in corpus (3)

• Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
• Asymptotic structure of self-shrinkers
• The space of asymptotically conical self-expanders of mean curvature flow

Cited by in corpus (10)

• Ancient gradient flows of elliptic functionals and Morse index
• Mean curvature flow with generic initial data
• Convexity of 2-convex translating and expanding solitons to the mean curvature flow in ${\mathbb{R}}^{n+1}$
• Initial Perturbation of the Mean Curvature Flow for closed limit shrinker
• Bounded Diameter Under Mean Curvature Flow
• Łojasiewicz-Simon inequalities for minimal networks: stability and convergence
• Łojasiewicz inequalities near simple bubble trees
• Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
• Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces
• Lectures on mean curvature flow of surfaces