中科院数学与系统科学研究院

数学研究所

偏微分方程研讨班

报告人：Prof. Li Yanyan (Rutgers University, USA)

题  目：Symmetry of hypersurface with ordered mean curvature in one direction

时  间：2019.11.03（星期日）, 10:30-11:30

地  点：数学院南楼N902室

摘  要：For a connected n-dimensional compact smooth hypersurface M without boundary embedded in R^{n+1}, a classical result of A.D. Aleksandrov shows that it must be a sphere if it has constant mean curvature. Nirenberg and I studied a one-directional analog of this result: if every pair of points (x', a), (x', b) in M with a < b has ordered mean curvature H(x', b) \le H(x', a), then M is symmetric about some hyperplane x_{n+1} = c under some additional conditions.Our proof was done by the moving plane method and some variations of the Hopf Lemma.  In a recent joint work with Xukai Yan and Yao Yao,we have obtained the symmetry of M under some weaker assumptions using a variational argument, giving a positive answer to a conjecture raised by Nirenberg and I in 2006.