Paper · Oct 6, 2017

A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold

Daguang Chen, Haizhong Li, Tailong Zhou

math.DGWider flowsarXiv:1710.02340
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Abstract

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordström-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which can be realized as graphs over Reissner-Nordström-anti-deSitter manifold.

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Submitted
Oct 6, 2017
Subjects
math.DG

Comments: 26 pages. All comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere (Frederico Girão and Neilha M. Pinheiro, 2016)3Girão 2016 Capacity inequalities and rigidity of cornered/conical manifolds (C. Tiarlos Cruz, 2017)4Cruz 2017 Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow (Brian Allen, 2017)8Allen 2017 Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space (Yingxiang Hu and Haizhong Li, 2018)2Hu 2018 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)1Andrews 2019 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)5Hu 2022 On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space (Brian Harvie, 2024)6Harvie 2024 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)7Pan 2024
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  1. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Haizhong Li · Shared terms “Alexandrov Fenchel”, “type inequality”, “Fenchel inequalities”

  2. Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Haizhong Li · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”, “type inequality”

  3. An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere

    Frederico Girão and Neilha M. PinheiroWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “type inequality”

  4. Capacity inequalities and rigidity of cornered/conical manifolds

    C. Tiarlos CruzWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Penrose type”, “Alexandrov Fenchel”, “type inequality”

  5. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Tailong Zhou · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”

  6. On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

    Brian HarvieWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Penrose type”, “Minkowski type”, “type inequality”

  7. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Minkowski type”, “type inequality”, “shaped hypersurfaces”

  8. Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “anti deSitter”, “Penrose”, “hyperbolic manifolds”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.