北航微分几何讨论班（第8讲）

题目：Contracting hypersurfaces by  powers of the $\sigma_k$-curvature

报告人：王险峰 （南开大学）

报告时间：2020年11月12日10:30-11:30

地点：沙河校区主楼E座E-404

摘要：We investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$  by $\sigma_k^\alpha$, where $\sigma_k$ is the $k$-th elementary symmetric function of the principal curvatures and $\alpha\ge 1/k$. We prove that for any $n\geq3$ and any fixed $k$ with $1\leq k\leq n$, there exists a constant  $c(n,k)>1/k$ such that  if $\alpha$ lies in the interval $[1/k,c(n,k)]$, then we  have a nice curvature pinching estimate involving the  ratio of the biggest principal curvature to the smallest principal curvature at every point of the flow hypersurface, and we prove that the properly rescaled  hypersurfaces converge exponentially to the unit sphere. Our results  provide an evidence for the general convergence result without initial curvature pinching conditions. This is joint work with Prof. Haizhong Li and Jing Wu.

报告人简介：王险峰，南开大学best365网页版登录，副教授。从事微分几何方向的研究，特别是子流形方面的研究。

邀请人：胡鹰翔、贺慧霞和张世金