论文标题
$ \ mathbb {r}^{3} $中平均曲率流的自相似解决方案
Self-similar solutions to the mean curvature flow in $\mathbb{R}^{3}$
论文作者
论文摘要
在本文中，我们通过革命表面和$ \ mathbb {r}^{3} $中的革命表面和规则表面对平均曲率流（MCF）进行分析。我们证明了MCF的自相似解决方案是通过$ \ Mathbb {r}^{3} $中的非圆柱形表面和圆锥形表面的自相似解决方案是微不足道的。此外，我们通过$ \ mathbb {r}^{3} $在生成曲线的曲率方面，在同一个螺旋运动下的旋转表面表征了MCF的自相似解决方案。最后，我们通过$ \ mathbb {r}^3 $在同一个螺旋运动下的圆柱体表面来表征MCF的自相似解决方案。还给出了$ \ mathbb {r}^{3} $在$ \ mathbb {r}^{3} $中通过圆柱体表面提供的精确解决方案的明确家族。
In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in $\mathbb{R}^{3}$ are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in $\mathbb{R}^{3}$ in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in $\mathbb{R}^3$. Explicit families of exact solutions for the MCF by cylindrical surfaces in $\mathbb{R}^{3}$ are also given.
