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

TL;DR

This paper uses numerical simulations to analyze the stability of Type-II singularities in mean curvature flow of noncompact hypersurfaces, showing that non-symmetric perturbations tend to develop similar singularities as symmetric ones.

Contribution

It extends previous stability analysis of Type-II singularities to non-symmetric perturbations using an adapted numerical method.

Findings

Non-symmetric perturbations near the tip lead to Type-II bowl soliton singularities.

Far from the tip, perturbations develop Type-I neckpinch singularities.

Angular dependence diminishes over time, leading to round singularities.

Abstract

In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that for small rotationally-symmetric perturbations of the initial embeddings that are sufficiently far away from the tip, MCF develops Type-I "neckpinch" singularities. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally-symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence -- becoming round -- and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this work is an adaptation of the "overlap" method introduced in [GIKW21] and permits angular dependence.