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November 2019 Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature
Siyuan Lu, Pengzi Miao
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J. Differential Geom. 113(3): 519-566 (November 2019). DOI: 10.4310/jdg/1573786973

Abstract

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam’s result is known to be equivalent to the Riemannian positive mass theorem.

In this paper, we provide a supplement to Shi–Tam’s result by including the boundary effect of minimal hypersurfaces. More precisely, given a compact manifold $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_H$ and $\Sigma_O$, where $\Sigma_H$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_O$ is assumed to be isometric to a suitable 2-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of mass $m$, we establish an inequality relating $m$, the area of $\Sigma_H$, and two weighted total mean curvatures of $\Sigma_O$ and $\Sigma$.

In $3$-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of $\Sigma_O$ being greater than or equal to the Hawking mass of $\Sigma_H$. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in an asymptotically flat $3$-manifold $M$ into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of $M$. It follows that our result on the compact manifold $\Omega$ is equivalent to the Riemannian Penrose inequality.

Funding Statement

The second-named author’s research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105.

Citation

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Siyuan Lu. Pengzi Miao. "Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature." J. Differential Geom. 113 (3) 519 - 566, November 2019. https://doi.org/10.4310/jdg/1573786973

Information

Received: 1 June 2017; Published: November 2019
First available in Project Euclid: 15 November 2019

zbMATH: 07130530
MathSciNet: MR4031741
Digital Object Identifier: 10.4310/jdg/1573786973

Rights: Copyright © 2019 Lehigh University

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Vol.113 • No. 3 • November 2019
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