On the precise asymptotics of Type-IIb solutions to mean curvature flow

2020/01/05 by Isenberg, James, Wu, Haotian, Zhang, Zhou
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.02123

Abstract

In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number γ>0, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as t\nearrow∞: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate (2t+1)^(γ-1)/2. (2) In a neighbourhood of the tip, the Type-IIb blow-up of the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface has a precise growth rate depending on γ.

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