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

IF 0.7 4区 数学 Q2 MATHEMATICS
David Garfinkle, James Isenberg, Dan Knopf, Haotian Wu
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引用次数: 0

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.
一类曲率放大的非紧超曲面平均曲率流的数值稳定性分析[j]
在之前的工作中,我们已经从数值模拟中提供了证据,证明由本文第二作者和第四作者构造的旋转对称、完全、非紧致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MCF受到非旋转对称的初始扰动的渐近行为,就像它在旋转对称扰动下的行为一样。特别是,如果我们对初始嵌入施加正弦角依赖,我们发现对于尖端附近的扰动,MCF的演化逐渐失去其角依赖-变得圆形并发展为ii型碗形孤子奇点。同样,如果我们对距离尖端足够远的扰动的初始嵌入施加正弦角依赖,那么随着i型颈夹奇点的发展,角依赖再次消失。本文进行的数值分析是对我们以前工作中引入的“重叠”方法的一种改编,并允许角相关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Experimental Mathematics
Experimental Mathematics 数学-数学
CiteScore
1.70
自引率
0.00%
发文量
23
审稿时长
>12 weeks
期刊介绍: Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.
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