Paper · Sep 30, 2016

Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

Siyuan Lu

math.DGWider flowsarXiv:1609.09733v2
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Abstract

In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.

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Submitted
Sep 30, 2016
Last revised
Jan 9, 2017 (v2)
Subjects
math.DG, math.AP

Comments: to appear in Comm. Anal. Geom

Other curvature flowsInverse mean curvature flowLong-time existence & convergencePhysics

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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 Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold (Li Chen and Jing Mao, 2016)1Chen 2016 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)2Scheuer 2017 Sobolev stability of the PMT and RPI using IMCF (Brian Allen, 2018)5Allen 2018 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)6Shi 2018 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)4Mao 2021 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)8Sun 2023 A locally constrained inverse Hessian quotient flow in de Sitter space (Kuicheng Ma, 2025)3Ma 2025 Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications (Fábio Nunes da Silva and Edwin Salinas Reyes, 2026)7da Silva 2026
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  1. Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

    Li Chen and Jing MaoWider flows

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

  2. 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”

  3. A locally constrained inverse Hessian quotient flow in de Sitter space

    Kuicheng MaWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Hessian quotient”, “inverse hessian”, “Sitter”

  4. 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 · Shared terms “star shaped”

  5. Sobolev stability of the PMT and RPI using IMCF

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Sitter Schwarzschild”

  6. 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 “Schwarzschild manifold”, “star shaped”

  7. Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications

    Fábio Nunes da Silva and Edwin Salinas ReyesWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Sitter”

  8. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows

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) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)