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

Abstract

In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number $\gamma >0$, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as $t\nearrow \infty$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate $(2t+1{)}^{(\gamma -1)/2}$. (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 $\gamma$.

Comments are welcome. arXiv admin note: text overlap with arXiv:1911.07282, arXiv:1603.01664