[Paper Review] Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Siyuan Lu, Pengzi Miao|arXiv (Cornell University)|Mar 23, 2017

Geometric Analysis and Curvature Flows32 references13 citations

TL;DR

This paper establishes a sharp inequality relating the mass of a spatial Schwarzschild manifold, the area of inner minimal hypersurfaces, and weighted total mean curvatures of outer boundary components in compact Riemannian manifolds with nonnegative scalar curvature. By incorporating minimal hypersurfaces and isometric embeddings into Schwarzschild space, the authors derive a quasi-local mass inequality that implies the Riemannian Penrose inequality and recovers the ADM mass in the asymptotic limit, extending the classical Shi-Tam theorem to include black hole horizon effects.

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.

Motivation & Objective

• To extend the Shi-Tam theorem by incorporating the presence of inner minimal hypersurfaces in compact manifolds with nonnegative scalar curvature.
• To establish a geometric inequality involving the mass of a Schwarzschild manifold, the area of minimal hypersurfaces, and weighted total mean curvatures of the outer boundary.
• To demonstrate that the resulting inequality is equivalent to the Riemannian Penrose inequality in 3 dimensions.
• To show that the limit of a quasi-local mass quantity associated with large coordinate spheres in an asymptotically flat 3-manifold equals the ADM mass, under suitable isometric embeddings into Schwarzschild space.

Proposed method

• The authors use the static potential function $ N = \frac{1 - \frac{m}{2}|x|^{1-n}}{1 + \frac{m}{2}|x|^{1-n}} $ on the spatial Schwarzschild manifold $ \mathbb{M}^{n+1}_m $ to weight the total mean curvature integrals on the boundary components.
• They derive a key inequality (1.1) involving the mass $ m $, the area of the inner minimal hypersurface $ \Sigma_H $, and weighted integrals of mean curvatures on $ \Sigma_O $ and its Schwarzschild model $ \Sigma $.
• The proof relies on the existence of isometric embeddings of the outer boundary $ \Sigma_O $ into a $ 2 $-convex hypersurface $ \Sigma \subset \mathbb{M}^{n+1}_m $, with $ \overline{\mathrm{Ric}}(\nu,\nu) \leq 0 $.
• For the asymptotic analysis, they consider large coordinate spheres $ S_r $ in an asymptotically flat 3-manifold $ M $, and compare their volume and mean curvature integrals to those in the model Schwarzschild space.
• They compute the difference $ V(r) - V_m(r) $ and show it asymptotically behaves like $ 2\pi(\mathfrak{m} - m)r^2 + o(r^2) $, linking the limit to the ADM mass $ \mathfrak{m} $.
• The equivalence to the Riemannian Penrose inequality is established via variational interpretation and limit analysis of the quasi-local mass functional.

Experimental results

Research questions

• RQ1 Can the Shi-Tam theorem be strengthened to include the contribution of inner minimal hypersurfaces in compact manifolds with nonnegative scalar curvature?
• RQ2 Is there a geometric inequality that relates the mass of a Schwarzschild manifold, the area of an inner horizon, and the total mean curvature of an outer boundary under isometric embedding?
• RQ3 Does the limit of a quasi-local mass quantity defined via isometric embeddings of large spheres into Schwarzschild space recover the ADM mass of the original manifold?
• RQ4 Is the derived inequality equivalent to the Riemannian Penrose inequality in 3 dimensions?

Key findings

• The paper establishes the inequality $ m + \frac{1}{n\omega_n}\int_\Sigma N H_m \, d\sigma \geq \frac{1}{2}\left(\frac{|\Sigma_H|}{\omega_n}\right)^{\frac{n-1}{n}} + \frac{1}{n\omega_n}\int_{\Sigma_O} N H \, d\sigma $, with equality if and only if $ \Sigma_H $ is nonempty and $ H = H_m $.
• In 3 dimensions, the inequality implies that the quasi-local mass of the outer boundary is at least the Hawking mass of the inner minimal surface.
• The limit of the quasi-local mass functional along large coordinate spheres in an asymptotically flat 3-manifold equals the ADM mass, as shown by $ \lim_{r\to\infty}\left(m + \frac{1}{8\pi}\int_{S_r} N(H_m - H)\,d\sigma\right) = \mathfrak{m} $.
• The volume difference $ V(r) - V_m(r) $ asymptotically satisfies $ V(r) - V_m(r) = 2\pi(\mathfrak{m} - m)r^2 + o(r^2) $, proving the equivalence to the Riemannian Penrose inequality.
• The result is equivalent to the Riemannian Penrose inequality in 3 dimensions, as confirmed by the limit behavior and variational interpretation of the functional on fill-ins.