A Remark on a Uniqueness Property of High Multiplicity Tangent Flows in Dimension Three

TL;DR

This paper establishes a uniqueness property for tangent flows at singularities in 3D mean curvature flow, showing that if one tangent flow is a multiple of a standard model, then all tangent flows are the same up to rotation.

Contribution

It combines previous work to prove that tangent flows at singular points are uniquely determined up to rotation when one tangent flow is a multiple of a standard model.

Findings

Uniqueness of tangent flows at singular points in 3D mean curvature flow.

All tangent flows are the same up to rotation if one is a multiple of a standard model.

The result applies to flows with tangent flows being a multiple of a plane, cylinder, or sphere.

Abstract

In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive integer multiple of a shrinking plane, cylinder or sphere, then, modulo rotations, all tangent flows at the point are the same.