Journal of Differential Geometry
- J. Differential Geom.
- Volume 113, Number 3 (2019), 519-566.
Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature
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.
The second-named author’s research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105.
J. Differential Geom., Volume 113, Number 3 (2019), 519-566.
Received: 1 June 2017
First available in Project Euclid: 15 November 2019
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Lu, Siyuan; Miao, Pengzi. Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature. J. Differential Geom. 113 (2019), no. 3, 519--566. doi:10.4310/jdg/1573786973. https://projecteuclid.org/euclid.jdg/1573786973