Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

2019/11/14 by James Isenberg, Haotian Wu, Isenberg, James +3
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1911.07282

Abstract

We continue the study, initiated by the first two authors in \citeIW19, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate (T-t)^-1. (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface approaches a collapsing cylinder at an exponential rate.

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