TY - JOUR

T1 - Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

AU - Lu, Siyuan

AU - Miao, Pengzi

N1 - Funding Information:
†The second named author’s research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105. Received June 1, 2017.

PY - 2019

Y1 - 2019

N2 - 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 boundary effect of minimal hypersurfaces. More precisely, given a compact manifold Ω with nonnegative scalar curvature, assuming its boundary consists of two parts, ΣH and ΣO, where ΣH is the union of all closed minimal hypersurfaces in Ω and ΣO is assumed to be isometric to a suitable 2-convex hypersurface Σ in a spatial Schwarzschild manifold of mass m, we establish an inequality relating m, the area of ΣH, and two weighted total mean curvatures of ΣO and Σ. In 3-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of ΣO being greater than or equal to the Hawking mass of ΣH. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in 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.

AB - 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 boundary effect of minimal hypersurfaces. More precisely, given a compact manifold Ω with nonnegative scalar curvature, assuming its boundary consists of two parts, ΣH and ΣO, where ΣH is the union of all closed minimal hypersurfaces in Ω and ΣO is assumed to be isometric to a suitable 2-convex hypersurface Σ in a spatial Schwarzschild manifold of mass m, we establish an inequality relating m, the area of ΣH, and two weighted total mean curvatures of ΣO and Σ. In 3-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of ΣO being greater than or equal to the Hawking mass of ΣH. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in 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.

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U2 - 10.4310/jdg/1573786973

DO - 10.4310/jdg/1573786973

M3 - Article

AN - SCOPUS:85079011422

VL - 113

SP - 519

EP - 566

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 3

ER -