Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

Abstract

Let $C$ be an $m$-dimensional cone immersed in ${\mathbb{R}}^{n+m}$. In this paper, we show that if $F:M^m\to {\mathbb{R}}^{n+m}$ is a properly immersed mean curvature flow self-shrinker which is smoothly asymptotic to $C$, then it is unique and converges to $C$ with unit multiplicity. Furthermore, if $F_1$ and $F_2$ are self-expanders that both converge to $C$ smoothly asymptotically and their separation decreases faster than ${\rho }^{-m-1}{e}^{-{\rho }^2/4}$ in the Hausdorff metric, then the images of $F_1$ and $F_2$ coincide.

44 pages. Exposition improved, several remarks added, various arguments corrected and expanded