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

TL;DR

This paper constructs rotationally symmetric solutions to mean curvature flow of noncompact hypersurfaces exhibiting Type-II curvature blow-up, with detailed asymptotics near the singularity, including convergence to a bowl soliton and cylindrical behavior at infinity.

Contribution

It provides explicit solutions with precise asymptotics for Type-II blow-up in mean curvature flow, extending previous work to include detailed near-singularity behavior.

Findings

Curvature concentrates at the tip and blows up at rate (T-t)^{-1}

Near the tip, solutions converge to the bowl soliton

At infinity, hypersurfaces approach a collapsing cylinder exponentially

Abstract

We continue the study, initiated by the first two authors in \cite{IW19}, 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.