Paper · Oct 4, 2016

Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

Li Chen, Jing Mao

math.DGWider flowsarXiv:1610.00836v3
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Abstract

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.

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Submitted
Oct 4, 2016
Last revised
Dec 18, 2016 (v3)
Subjects
math.DG
Journal
A slight different version was published in J. Geom. Anal. 28 (2018) 921-949
DOI
10.1007/s12220-017-9848-6

Comments: 22 pages. Some revisions have been made to v2

Other curvature flowsInverse mean curvature flowHomogeneous spacesLong-time existence & convergencePhysics

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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 Inverse curvature flow in anti-de Sitter-Schwarzschild manifold (Siyuan Lu, 2016)1Lu 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)2Chen 2017 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)7Scheuer 2017 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)8Shi 2018 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)3Mao 2021 Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)5Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)6Gao 2021 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)4Gao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

    Siyuan LuWider flows

    Topics Inverse mean curvature flow, Physics, Long-time existence & convergence · Shared terms “Schwarzschild manifold”, “Sitter Schwarzschild”, “shaped initial”

  2. Inverse mean curvature flow inside a cone in warped products

    Li Chen, Jing Mao, Ni Xiang et al.Wider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same authors Li Chen, Jing Mao · Shared terms “hypersurfaces converge”, “star shaped”

  3. A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

    Jing Mao and Qiang TuWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “star shaped”, “hypersurfaces converge”

  4. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “hypersurfaces converge”, “star shaped”

  5. Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Physics, Long-time existence & convergence · Same author Jing Mao · Shared terms “hypersurfaces converge”

  6. An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Physics, Long-time existence & convergence · Same author Jing Mao · Shared terms “hypersurfaces converge”

  7. Locally constrained inverse curvature flows

    Julian Scheuer and Chao XiaWider flows

    Topics Inverse mean curvature flow, Physics, Long-time existence & convergence · Shared terms “Schwarzschild manifold”, “Sitter Schwarzschild”

  8. Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension

    Yuguang Shi and Jintian ZhuWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “AdS Schwarzschild”, “Schwarzschild manifold”, “star shaped”

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.

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.