# MATH Seminar

Title: Symmetry of hypersurfaces with ordered mean curvature in one direction
Colloquium: Analysis and Differential Geometry
Speaker: Yanyan Li of Rutgers University
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.