Backwards uniqueness of the mean curvature flow

TL;DR

This paper proves that if two complete hypersurfaces evolving under mean curvature flow in Euclidean space coincide at a final time, then they must have been identical throughout the entire flow, establishing a backwards uniqueness property.

Contribution

It establishes the backwards uniqueness of the mean curvature flow for hypersurfaces with bounded second fundamental forms, extending analogous results from Ricci flow.

Findings

Uniqueness of solutions given identical final states

Applicability to hypersurfaces with bounded second fundamental forms

Extension of Ricci flow uniqueness results

Abstract

In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the mean curvature flow on $M^n \times [0,T]$ with bounded second fundamental forms. Suppose $F_T=\widetilde{F}_T$, then $F_t=\widetilde{F}_t$ on $M^n \times [0,T]$. This is an analog of a result of Kotschwar on the Ricci flow.