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

2020/05/15 by Khan, Ilyas · 1 citation
#35J15 (Secondary) #53C44 (Primary) 53C24 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.07611

Abstract

Let C be an m-dimensional cone immersed in ℝ^n+m. In this paper, we show that if F:M^m → 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 F₁ and F₂ are self-expanders that both converge to C smoothly asymptotically and their separation decreases faster than ρ^-m-1e^-ρ2/4 in the Hausdorff metric, then the images of F₁ and F₂ coincide.

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• Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

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