Open Access
October 2013 Hypersurfaces with nonnegative scalar curvature
Lan-Hsuan Huang, Damin Wu
J. Differential Geom. 95(2): 249-278 (October 2013). DOI: 10.4310/jdg/1376053447


We show that closed hypersurfaces in Euclidean space with non-negative 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 counterexamples for all $k$ greater than 2. Our proof relies on a new geometric argument 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 noncompact asymptotically flat hypersurfaces with non-negative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.


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Lan-Hsuan Huang. Damin Wu. "Hypersurfaces with nonnegative scalar curvature." J. Differential Geom. 95 (2) 249 - 278, October 2013.


Published: October 2013
First available in Project Euclid: 9 August 2013

zbMATH: 1279.53009
MathSciNet: MR3128984
Digital Object Identifier: 10.4310/jdg/1376053447

Rights: Copyright © 2013 Lehigh University

Vol.95 • No. 2 • October 2013
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