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 talk, 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 with nonnegative scalar curvature, assuming its boundary consists of two parts, horizon boundary and outer boundary, where horizon boundary is the union of all closed minimal hypersurfaces and outer boundary is isometric to a suitable 2-convex hypersurface in a spatial Schwarzschild manifold of positive mass m, we establish an inequality relating m, the area of horizon boundary , and two weighted total mean curvatures of the outer boundary and the hypersurface in the Schwarzschild manifold. In 3-dimension, this inequality is equivalent to the Riemannian Penrose inequality. This talk is based on joint work with Pengzi Miao.