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.
Publisher Copyright:
© 2019 International Press of Boston, Inc.. All rights reserved.
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 -