# Lan-Hsuan Huang : Hypersurfaces with nonnegative scalar curvature and a positive mass theorem

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

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