Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

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 effect of minimal hypersurfaces on the boundary. 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 isometric to a suitable $2$-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of positive 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, the inequality has implications to both isometric embedding and quasi-local mass problems. In a relativistic context, our 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 such quasi-local mass quantity associated with suitably chosen isometric embeddings of large coordinate spheres of 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.

Introduction revised, references updated, 43 pages

References in corpus (5)

• Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold
• The Weyl problem in warped product space
• A volume preserving flow and the isoperimetric problem in warped product spaces
• Quasi-local energy with respect to a static spacetime
• Higher dimensional black hole initial data with prescribed boundary metric

Cited by in corpus (8)

• Inverse curvature flows in Riemannian warped products
• On Weyl's embedding problem in Riemannian manifolds
• On the Rigidity of Riemannian-Penrose Inequality for Asymptotically Flat 3-manifolds with Corners
• Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture
• A rigidity theorem for surfaces in Schwarzschild manifold
• Variation and rigidity of quasi-local mass
• Bartnik mass via vacuum extensions
• Uniqueness of isometric immersions with the same mean curvature