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 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.

Geometric Inequalities for Quasi-Local Masses

“…In a similar manner, by combining the proofs of Theorem 2.10 above and Theorem 1.1 in [43] we obtain the following result. Theorem 8.2.…”

“…Proof. This result follows by combining the proof of Theorem 2.8 above, and the proof of Theorem 1.1 in[43]. Here we provide an outline.…”

“…where H is the mean curvature of the embedding Σ ֒→ Ω. Recently, Lu and Miao [43] have proven a Penrose type inequality for the static Brown-York mass in which the reference static spacetime is the Schwarzschild solution. Here we give an extension of their result to the static Liu-Yau mass setting.…”

“…Here we show how to smooth the gluing construction to allow for a conformal change to nonnegative scalar curvature, and thus avoid having to use spinors to establish positive mass. Lastly, we will also generalize a result of Lu-Miao [43] to obtain a localized version of the Penrose inequality for the static Wang-Yau mass [15]. In the remainder of the paper geometrized units will be used so that c = G = 1.…”

A quasi-local Penrose inequality for the quasi-local energy with static references

“…The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with reference in any spherically symmetric static spacetime.…”