Singularities of the mean curvature flow of an embedded surface in 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.
"Uniqueness of asymptotically conical tangent flows." Duke Math. J. 170 (16) 3601 - 3657, 1 November 2021. https://doi.org/10.1215/00127094-2020-0098