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Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature
Journal of Differential Geometry ( IF 2.5 ) Pub Date : 2019-11-01 , DOI: 10.4310/jdg/1573786973
Siyuan Lu 1 , Pengzi Miao 2
Affiliation  

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 $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_h$ and $\Sigma_o$, where $\Sigma_h$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_o$ is isometric to a suitable $2$-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of positive mass $m$, we establish an inequality relating $m$, the area of $\Sigma_h$, and two weighted total mean curvatures of $\Sigma_o$ and $ \Sigma$. 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 $ \Sigma_o$ being greater than or equal to the Hawking mass of $\Sigma_h$. 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 $\Omega$ is equivalent to the Riemannian Penrose inequality.

中文翻译:

具有非负标量曲率的致密流形的最小超曲面和边界行为

在边界具有正平均曲率的紧黎曼流形上,Shi 和 Tam 的基本结果表明,如果流形具有非负标量曲率,并且如果边界与欧几里德空间中的严格凸超曲面等距,则总平均曲率边界的总平均曲率不大于相应欧几里得超曲面的总平均曲率。在 $3$ 维度中,Shi-Tam 的结果已知等同于黎曼正质量定理。在本文中,我们通过包括最小超曲面对边界的影响来补充 Shi-Tam 的结果。更准确地说,给定一个具有非负标量曲率的紧凑流形 $\Omega$,假设它的边界由两部分组成,$\Sigma_h$ 和 $\Sigma_o$,其中,$\Sigma_h$ 是 $\Omega$ 中所有闭合的最小超曲面的并集,并且 $\Sigma_o$ 与正质量 $m$ 的空间 Schwarzschild 流形中合适的 $2$-凸面超曲面 $\Sigma$ 等距,我们建立与 $m$、$\Sigma_h$ 的面积以及 $\Sigma_o$ 和 $\Sigma$ 的两个加权总平均曲率相关的不等式。在 $3$ 维度中,不等式对等距嵌入和准局部质量问题都有影响。在相对论的背景下,我们的结果可以解释为 $\Sigma_o$ 的准局部质量类型量大于或等于 $\Sigma_h$ 的霍金质量。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. 我们表明极限等于 $M$ 的 ADM 质量。因此,我们在紧流形 $\Omega$ 上的结果等价于黎曼彭罗斯不等式。
更新日期:2019-11-01
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