Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

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