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 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 isometric to a suitable 2-convex hypersurface Σ in a spatial Schwarzschild manifold of positive 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, 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 Σ O being greater than or equal to the Hawking mass of Σ 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 Ω is equivalent to the Riemannian Penrose inequality.

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds

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We consider a priori estimates of Weyl's embedding problem of (S 2 , g) in general 3-dimensional Riemannian manifold (N 3 ,ḡ). We establish interior C 2 estimate under natural geometric assumption. Together with a recent work by Li and Wang, we obtain an isometric embedding of (S 2 , g) in Riemannian manifold. In addition, we reprove Weyl's embedding theorem in space form under the condition that g ∈ C 2 with D 2 g Dini continuous.

We consider the exterior Dirichlet problem for Monge-Ampère equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and the first named author, we complete the characterization of the existence and nonexistence of solutions in terms of their asymptotic behaviors.