Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
Speaker:
Prof. Li Yanyan，Rutgers University, USA
Inviter:
Title:
Symmetry of hypersurface with ordered mean curvature in one direction
Time & Venue:
2019.11.03 10:30-11:30 N902
Abstract:
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.
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