Self-similar solutions of $\sigma_k^{\alpha}$-curvature flow

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

Differential Geometry mathscidoc:1611.10006

2016.11
In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $\sigma_k^{\alpha}$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $F$. At last, we prove that if $F$ can be compared with $\frac{(n-k+1)\sigma_{k-1}}{k\sigma_{k}}$, then a closed strictly $k$-convex solution of $(*)$ must be a round sphere.
$\sigma_k$ curvature, self-similar solution, non-homogeneous curvature function
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@inproceedings{shanze2016self-similar,
  title={Self-similar solutions of $\sigma_k^{\alpha}$-curvature flow},
  author={Shanze Gao, and Hui Ma},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161123100557713732625},
  year={2016},
}
Shanze Gao, and Hui Ma. Self-similar solutions of $\sigma_k^{\alpha}$-curvature flow. 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161123100557713732625.
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