Asymptotic convergence for modified scalar curvature flow
Asymptotic convergence for modified scalar curvature flow
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DOI：
10.4310/cag.2023.v31.n1.a3
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发表时间：
2020-08
影响因子：
0.7
通讯作者：
Ling Xiao
中科院分区：
文献类型：
--
作者：
Ling Xiao
本文研究了$\mathbb{R}^{n+1}$中速度为$r^\alpha\sigma_2^{1/2}$的闭星形超曲面的流，$其中$\sigma_2 ^{1/2}$是标量曲率的归一化平方根，$\alpha\geq 2，$和$r$是超曲面上的点到原点的距离。我们证明了流是永远存在的，并且星形性是保持的。此外，归一化后，我们表明，流量以指数速度收敛到一个以原点为中心的球体。当$\alpha<2时，给出了上述收敛性的一个反例。
In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha\sigma_2^{1/2},$ where $\sigma_2^{1/2}$ is the normalized square root of the scalar curvature, $\alpha\geq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $\alpha<2,$ a counterexample is given for the above convergence.