A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

TL;DR

This paper numerically investigates the stability of mean curvature flow in noncompact hypersurfaces with Type-II blowup, revealing distinct behaviors based on initial data and proposing a critical class leading to degenerate neckpinches.

Contribution

It introduces a novel overlap numerical method to analyze stability and identifies classes of initial data leading to different singular behaviors in mean curvature flow.

Findings

Existence of two classes of initial data with distinct singular behaviors.

Numerical evidence for a critical class leading to degenerate neckpinches.

Confirmation that curvature blowup rates can vary under perturbations.

Abstract

We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a Type-II rate no slower than $(T-t)^{-1}$. Secondly, there is a "far" class of initial data which lead to solutions developing a local Type-I nondegenerate neckpinch under mean curvature flow. These numerical findings further suggest the existence of a "critical" class of initial data which conjecturally lead to mean curvature flow of noncompact hypersurfaces forming local Type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than $(T-t)^{-1}$.