Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

TL;DR

This paper proves a backwards uniqueness result for mean curvature flow with isolated asymptotically conical singularities, establishing that identical singularities imply identical flows without assuming self-shrinking or global asymptotic behavior.

Contribution

It introduces the first backwards uniqueness theorem for geometric flows with singularities, developing new tools to handle singularity cores and asymptotic structures.

Findings

Backward uniqueness holds for mean curvature flows with isolated asymptotically conical singularities.

Low entropy flows in -dimensional space are uniquely determined by their singularity at the first singular time.

New global analytical tools are developed to manage singularity structures in geometric flows.

Abstract

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces $M^1_t$ and $M^2_t$ encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time $T$, and if $M^1_T=M^2_T$ then $M^1_t=M^2_t$ for every $t\in [0,T]$. This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in $\mathbb{R}^4$ are backwards unique