Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow

Shanze Gao Tsinghua University, Beijing Haizhong Li Tsinghua University, Beijing Hui Ma Tsinghua University, Beijing

Differential Geometry mathscidoc:1701.10018

2017.1
By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $\sigma_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with $\alpha\geq \frac{1}{k}$, must be a round sphere.
$\sigma_k$ curvature, self-similar solution
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@inproceedings{shanze2017uniqueness,
  title={Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow},
  author={Shanze Gao, Haizhong Li, and Hui Ma},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170111143232336904098},
  year={2017},
}
Shanze Gao, Haizhong Li, and Hui Ma. Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow. 2017. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170111143232336904098.
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