Mean curvature rigidity phenomenon and its extensions

：马翔（北京大学）
：2022-03-25 14:15
：腾讯会议ID：149127155（无密码）

报告人：马翔（北京大学）

时  间：3月25日下午14:15

地  点：腾讯会议ID：149127155（（无密码）

内容摘要:

A theorem by Gromov asserts that for a hyperplane in the Euclidean space E^n, any smooth perturbation with compact support and nonnegative mean curvature H must be trivial (i.e. identical to the original one). We will start by presenting Souam's simple proof of this rigidity result using the tangency principle. Then we consider similar problems for the unit (hyper-)sphere with mean curvature H=1 in E^n. Our main result says that when one perturbs the sphere only in a hemisphere, and the mean curvature H is no less than 1 for this smooth hypersurface after perturbation, then under quite natural conditions it must be congruent to the round sphere. On the other hand, if the fixed part of the sphere is only a small spherical cap, then there exist nontrivial perturbations on the complementary great spherical cap such that H is greater than 1 on the perturbed part. This is a joint work with Prof. Shibing CHEN (from USTC) and my student Shengyang WANG (from PKU).

个人简介：

马翔，北京大学数学科学学院教授、博士生导师，数学系副主任。2005年在德国柏林工业大学获得博士学位，此后一直任教于北京大学。专长微分几何，兴趣集中于子流形的整体几何，尤其是极小曲面、Willmore曲面和几何不变量等主题。在Journal of Differential Geometry、Advances in Mathematics等期刊发表论文约20篇。

 

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