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

TL;DR

This paper investigates Type-II curvature blow-up in mean curvature flows of noncompact, rotationally symmetric hypersurfaces, revealing detailed asymptotic behaviors and convergence properties near singularities.

Contribution

It introduces analytic techniques using matched asymptotics and barrier solutions to analyze singularity formation and describes the asymptotic profile near the blow-up time.

Findings

Curvature concentrates at the hypersurface tip with blow-up rate depending on parameter γ.

Near the tip, solutions converge to a higher-dimensional Grim Reaper translating soliton.

Away from the tip, the surface approaches a collapsing cylinder at a rate influenced by γ.

Abstract

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $\gamma>1/2$, there is a solution with the highest curvature blowing up at the rate $(T-t)^{-(\gamma +1/2)}$. (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter $\gamma$.