Hypersurfaces with nonnegative scalar curvature

Lan-Hsuan Huang University of Connecticut Damin Wu University of Connecticut

Differential Geometry mathscidoc:1609.10331

Distinguished Paper Award in 2017

Journal of Differential Geometry, 95, (2), 249-278, 2013
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 kth 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 nonnegative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.
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@inproceedings{lan-hsuan2013hypersurfaces,
  title={HYPERSURFACES WITH NONNEGATIVE SCALAR CURVATURE},
  author={Lan-Hsuan Huang, and Damin Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914082258913659993},
  booktitle={Journal of Differential Geometry},
  volume={95},
  number={2},
  pages={249-278},
  year={2013},
}
Lan-Hsuan Huang, and Damin Wu. HYPERSURFACES WITH NONNEGATIVE SCALAR CURVATURE. 2013. Vol. 95. In Journal of Differential Geometry. pp.249-278. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914082258913659993.
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