Paper · Jan 11, 2018

Conformal Ricci flow on asymptotically hyperbolic manifolds

Peng Lu, Jie Qing, Yu Zheng

math.DGarXiv:1801.03869
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Abstract

In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.

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Submitted
Jan 11, 2018
Subjects
math.DG

Comments: 19 pages. No figure

Ricci flowBoundaries & non-compactHarmonic maps & DeTurck trick

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 New proofs of Perelman's theorem on shrinking Breathers in Ricci flow (Peng Lu and Yu Zheng, 2017)5Lu 2017 Asymptotically Hyperbolic 3-Metric with Ricci flow foliation (Hyun Chul Jang, 2018)7Jang 2018 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)8Shi 2018 Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci (Mahdi Ammar, 2019)6Ammar 2019 Monotonicity of functionals along conformal Ricci flow (Fengjiang Li, Peng Lu, Jianhong Wang et al., 2019)1Li 2019 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)3Huang 2021 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)4Bahuaud 2023 A volume-renormalized mass for asymptotically hyperbolic manifolds (Mattias Dahl, Klaus Kroencke and Stephen McCormick, 2023)2Dahl 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Monotonicity of functionals along conformal Ricci flow

    Fengjiang Li, Peng Lu, Jianhong Wang et al.

    Topics Ricci flow · Same authors Peng Lu, Yu Zheng · Shared terms “conformal Ricci”

  2. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick

    Topics Boundaries & non-compact · Shared terms “hyperbolic manifolds”, “asymptotically hyperbolic”

  3. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “short time”, “time existence”

  4. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al.

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “asymptotically hyperbolic”

  5. New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

    Peng Lu and Yu Zheng

    Topics Ricci flow · Same authors Peng Lu, Yu Zheng

  6. Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci

    Mahdi Ammar

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact

  7. Asymptotically Hyperbolic 3-Metric with Ricci flow foliation

    Hyun Chul Jang

    Topics Boundaries & non-compact · Shared terms “asymptotically hyperbolic”

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

    Yuguang Shi and Jintian ZhuWider flows

    Topics Boundaries & non-compact · Shared terms “hyperbolic manifolds”, “asymptotically hyperbolic”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.