论文标题
HOPF类型定理用于曲率流的自相似解决方案$ \ Mathbb {r}^3 $
Hopf type theorems for self-similar solutions of curvature flows in $\mathbb{R}^3$
论文作者
论文摘要
在本文中，我们证明了二维，封闭，沉浸，非必要的凸，相似的解决方案的刚性结果，以$ \ mathbb {r}^3 $中的一系列完全非线性抛物线流的范围。我们显示，这种自相似的溶液是以原点为中心的圆形球体，只要它具有零属，并且满足了高斯曲率的合适的上夹克估计值。作为应用，我们获得了圆球的刚性结果，作为唯一的封闭，浸泡，零属的几个已知流动的自相似解，作为平均曲率的能力，谐波平均曲率流量和$α$ -GAUSSIAN曲率流量的流动，以$ $α\ in（0,1/4）$。我们指出，我们的结果不假定任何嵌入性条件。
In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the $α$-Gaussian curvature flow for $α\in(0,1/4)$. We remark that our result does not assume any embeddedness condition.
