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 Ω with nonnegative scalar curvature, assuming its boundary consists of two parts, Σ_H and Σ_O, where Σ_H is the union of all closed minimal hypersurfaces in Ω and Σ_O is assumed to be isometric to a suitable 2-convex hypersurface Σ in a spatial Schwarzschild manifold of mass m, we establish an inequality relating m, the area of Σ_H, and two weighted total mean curvatures of Σ_O and Σ. 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 Σ_O being greater than or equal to the Hawking mass of Σ_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 Ω is equivalent to the Riemannian Penrose inequality.