Paper · Feb 3, 2018

Asymptotically Hyperbolic 3-Metric with Ricci flow foliation

Hyun Chul Jang

math.DGarXiv:1802.01019
View PDF View HTML arXiv page

As on arXiv

Abstract

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, C. -Y. Lin used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension and C. Sormani and Lin proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass.

View PDF View HTML arXiv page
Submitted
Feb 3, 2018
Subjects
math.DG
DOI
10.1007/s00023-018-0745-8
Ricci flowBoundaries & non-compactPhysics

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 Ricci Flow and Volume Renormalizability (Eric Bahuaud, Rafe Mazzeo and Eric Woolgar, 2016)5Bahuaud 2016 Conformal Ricci flow on asymptotically hyperbolic manifolds (Peng Lu, Jie Qing and Yu Zheng, 2018)6Lu 2018 RG-2 flow, mass and entropy (Oscar Lasso Andino, 2018)3Andino 2018 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)2Shi 2018 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)8Burkhardt-Gu 2022 A volume-renormalized mass for asymptotically hyperbolic manifolds (Mattias Dahl, Klaus Kroencke and Stephen McCormick, 2023)1Dahl 2023 Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)7Gui 2024 A continuous Positive Mass Theorem for perturbations of Euclidean space (Paula Burkhardt-Guim, 2026)4Burkhardt-Gu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick

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

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

    Yuguang Shi and Jintian ZhuWider flows

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically hyperbolic”, “total mass”, “asymptotically flat”

  3. RG-2 flow, mass and entropy

    Oscar Lasso Andino

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically flat”, “Hawking mass”, “foliation”

  4. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically flat”

  5. Ricci Flow and Volume Renormalizability

    Eric Bahuaud, Rafe Mazzeo and Eric Woolgar

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

  6. Conformal Ricci flow on asymptotically hyperbolic manifolds

    Peng Lu, Jie Qing and Yu Zheng

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

  7. Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces

    Yaoting Gui, Yuqiao Li and Jun SunWider flows

    Topics Physics, Boundaries & non-compact · Shared terms “foliation”, “asymptotically flat”

  8. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically flat”

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) 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)

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.