Siyuan Lu, Pengzi Miao

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.