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.

摘要

在之前的工作中，我们通过数值模拟提供了证据，表明由本文第二和第四作者构造的旋转对称、完整、非紧嵌入超曲面的平均曲率流（MCF）的 II 型奇点是稳定的。在这项工作中，我们再次使用数值模拟来表明，受到非旋转对称初始扰动的 MCF 的渐近行为就像旋转对称扰动一样。特别是，如果我们对初始嵌入施加正弦角度依赖性，我们会发现对于尖端附近的扰动，MCF 的演化会逐渐失去其角度依赖性（变成圆形）并形成 II 型碗孤子奇点。还有，如果我们对距尖端足够远的扰动的初始嵌入施加正弦角度依赖性，则随着 I 型颈缩奇点的发展，角度依赖性再次消失。本文中进行的数值分析是我们之前工作中引入的“重叠”方法的改编，并且允许角度依赖性。