Hypersurfaces with nonnegative scalar curvature

Abstract

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.

A point in the proof of Theorem 2 that was overlooked in the previous versions is fixed. The appendix of some topological results is added. To appear in J. Differential. Geom

References in corpus (5)

• The Higher Dimensional Positive Mass Theorem II
• On the Positive Mass, Penrose, an ZAS Inequalities in General Dimension
• Rigidity phenomena involving scalar curvature
• Geometric inequalities and rigidity theorems on equatorial spheres
• The equality case of the Penrose inequality for asymptotically flat graphs

Cited by in corpus (1)

• The equality case of the Penrose inequality for asymptotically flat graphs