Paper · Jul 24, 2018

The Exponential Convergence of the CR Yamabe Flow

Weimin Sheng, Kunbo Wang

math.DGWider flowsarXiv:1807.08876
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Abstract

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

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Submitted
Jul 24, 2018
Subjects
math.DG

Comments: Accepted for publication in SCIENCE CHINA Mathematics

Yamabe flowScalar curvatureLong-time existence & convergence

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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)5Bahuaud 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)2Amacha 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)1Ho 2017 Yamabe flow on non-compact manifolds with unbounded initial curvature (Mario B. Schulz, 2018)7Schulz 2018 CR Yamabe constant, CR Yamabe flow and its soliton (Pak Tung Ho and Kunbo Wang, 2020)4Ho 2020 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)8Yan 2021 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)3Lye 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow · Same authors Weimin Sheng, Kunbo Wang

  2. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence

  3. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence

  4. CR Yamabe constant, CR Yamabe flow and its soliton

    Pak Tung Ho and Kunbo WangWider flows

    Topics Yamabe flow · Same author Kunbo Wang

  5. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Shared terms “Yamabe invariant”

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

    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow, Scalar curvature

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

    Mario B. SchulzWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence · Shared terms “Yamabe invariant”

  8. Convergence of the Weighted Yamabe Flow

    Zetian YanWider flows

    Topics Yamabe flow, Scalar curvature, Long-time existence & convergence

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) 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) 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. 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.