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

Abstract

We continue the study, initiated by the first two authors in [15], of Type-II curvature blow-up in mean curvature flow of complete noncompact hypersurfaces embedded in Euclidean space. 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.