Speaker：Xiang Ma（Peking University）

Time：2022-03-25 14:15

Location：Tencent Meeting ID：149127155（No Password）

Abstract: 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).