论文标题
具有II型曲率爆炸的非脉冲超曲面的平均曲率流量的数值稳定性分析：II
A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II
论文作者
论文摘要
在先前的工作[GIKW21]中，我们从数值模拟中提供了证据，表明旋转对称，完整，完全，非脉冲嵌入的超丘的平均曲率流量（MCF）的类型奇异性是[IW19，IWZ21]中构建的。更准确地说，在该论文中显示，对于“尖端”附近的初始嵌入的小旋转对称性扰动，此类初始嵌入的MCF的数值模拟在奇异的社区中具有相同的“ Bowl soliton”爆炸行为相同。在这项工作中还显示，对于距尖端足够远的初始嵌入的小旋转对称扰动，MCF会形成I型“颈钉”奇点。 在这项工作中，我们再次使用数值模拟表明，MCF受到不旋转对称性的初始扰动的影响，就像旋转对称的扰动一样渐近地行为。特别是，如果我们对初始嵌入施加正弦角的依赖性，我们发现，对于尖端附近的扰动，MCF渐近地逐渐失去了角度的依赖性 - 变得圆形 - 圆形 - II型碗soliton奇异性。同样，如果我们对足够远离尖端的扰动的初始嵌入施加正弦角依赖性，则随着I型i颈刺的发展，角度依赖性再次消失。在这项工作中进行的数值分析是对[GIKW21]中引入的“重叠”方法的适应，并允许角度依赖性。
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.
