A Numerical Stability Analysis of Mean Curvature Flow of Noncompact Hypersurfaces with Type-II Curvature Blowup: II

Abstract

In previous work, 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 by the second and the fourth authors of this paper, are stable. 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 paper is an adaptation of the “overlap” method introduced in our previous work and permits angular dependence.