Lan-Hsuan Huang : Hypersurfaces with nonnegative scalar curvature and a positive mass theorem (Sep 21, 2011 4:25 PM)

Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite. In a joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that closed hypersurfaces with non-negative scalar curvature must be weakly mean convex. The proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of the hypersurface to the geometry of the level sets of a height function. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.

• Category: Geometry and Topology
• Duration: 01:34:52
• Date:  September 21, 2011 at 4:25 PM
• Views: 121
• Tags:   seminar, Geometry/Topology Seminar