Hypersurfaces with nonnegative scalar curvature

Details Hypersurfaces with nonnegative scalar curvature Hypersurfaces with nonnegative scalar curvature

2011-02-28

arxiv.org/abs/1102.5749v2

Mathematics

Differential Geometry

25 pages, 1 figure. Main theorems are strengthened

Scientific paper

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete non-compact hypersurfaces of finitely many regular ends with nonnegative scalar curvature are weakly mean convex, and prove a positive mass theorem for such hypersurfaces.

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