Uniqueness of asymptotically conical higher codimension self-shrinkers and self-expanders

Abstract

Abstract Let 𝐶 be an 𝑚-dimensional cone immersed in R n + m \mathbb{R}^{n+m} . In this paper, we show that if F : M m → R n + m F\colon M^{m}\rightarrow\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 F_{1} and F 2 F_{2} are self-expanders that both converge to 𝐶 smoothly asymptotically and their separation decreases faster than ρ − m − 1 ⁢ e − ρ 2 / 4 \rho^{-m-1}e^{-\rho^{2}/4} in the Hausdorff metric, then the images of F 1 F_{1} and F 2 F_{2} coincide.