2019/01/31 by Otis Chodosh, Felix Schulze · 2 citations
Mathematics · #math.AP #math.DG
paper · pdf · doi:10.1215/00127094-2020-0098
40 pages. Final version, to appear in Duke Mathematical Journal
arxiv created 2020/12/16 · arxiv updated 2021/12/06
Singularities of the mean curvature flow of an embedded surface in R3 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.