Paper · Feb 4, 2021

Global Yamabe flow on asymptotically flat manifolds

Li Ma

math.DGWider flowsarXiv:2102.02399
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Abstract

In this paper, we study the existence of global Yamabe flow on asymptotically flat (in short, AF or ALE) manifolds. Note that the ADM mass is preserved in dimensions 3,4 and 5. We present a new general local existence of Yamabe flow on a complete Riemannian manifold with the initial metric quasi-isometric to a background metric of bounded scalar curvature. Asymptotic behaviour of the Yamabe flow on ALE manifolds is also addressed provided the initial scalar curvature is non-negative and there is a bounded subsolution to the corresponding Poisson equation. We also present a maximum principle for a very general parabolic equations on the complete Riemannian manifolds.

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Submitted
Feb 4, 2021
Subjects
math.DG

Comments: 20 pages, submitted

Yamabe flowRicci-flat & Calabi–YauScalar curvatureBoundaries & 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 Convergence of the Yamabe flow on manifolds with minimal boundary (Sergio Almaraz and Liming Sun, 2016)6Almaraz 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)7Amacha 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)8Ho 2017 Yamabe flow on non-compact manifolds with unbounded initial curvature (Mario B. Schulz, 2018)4Schulz 2018 Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability (Li Ma, 2020)2Ma 2020 The Yamabe flow on asymptotically flat manifolds (Eric Chen and Yi Wang, 2021)1Chen 2021 Normalized Yamabe flow on manifolds with bounded geometry (Bruno Caldeira, Luiz Hartmann and Boris Vertman, 2021)5Caldeira 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)3Carron 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Yamabe flow on asymptotically flat manifolds

    Eric Chen and Yi WangWider flows

    Topics Yamabe flow, Physics, Scalar curvature · Shared terms “asymptotically flat”, “flat manifolds”, “ADM mass”

  2. Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability

    Li MaWider flows

    Topics Yamabe flow, Scalar curvature · Same author Li Ma · Shared terms “complete Riemannian”, “initial metric”, “Riemannian manifold”

  3. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow, Boundaries & non-compact · Shared terms “asymptotically flat”, “flat manifolds”

  4. Yamabe flow on non-compact manifolds with unbounded initial curvature

    Mario B. SchulzWider flows

    Topics Yamabe flow, Scalar curvature, Boundaries & non-compact · Shared terms “background metric”, “initial metric”

  5. Normalized Yamabe flow on manifolds with bounded geometry

    Bruno Caldeira, Luiz Hartmann and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature, Boundaries & non-compact · Shared terms “complete Riemannian”, “Riemannian manifold”

  6. Convergence of the Yamabe flow on manifolds with minimal boundary

    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “Riemannian manifold”

  7. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “Riemannian manifold”

  8. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.