## Journal of Differential Geometry

### Hypersurfaces with nonnegative scalar curvature

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 95, Number 2 (2013), 249-278.

Dates
First available in Project Euclid: 9 August 2013

https://projecteuclid.org/euclid.jdg/1376053447

Digital Object Identifier
doi:10.4310/jdg/1376053447

Mathematical Reviews number (MathSciNet)
MR3128984

Zentralblatt MATH identifier
1279.53009

#### Citation

Huang, Lan-Hsuan; Wu, Damin. Hypersurfaces with nonnegative scalar curvature. J. Differential Geom. 95 (2013), no. 2, 249--278. doi:10.4310/jdg/1376053447. https://projecteuclid.org/euclid.jdg/1376053447