Paper · Jul 6, 2021

Convergence rate of the prescribed curvature flow

Pak Tung Ho, Jinwoo Shin

math.DGWider flowsarXiv:2107.09616v2
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Abstract

The prescribed scalar curvature flow was introduced to study the problem of prescribing scalar curvature on manifolds. Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study in this paper the convergence rate of the prescribed scalar curvature flow.

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Submitted
Jul 6, 2021
Last revised
May 4, 2023 (v2)
Subjects
math.DG

Comments: Example in section 5 is not correct

Yamabe flowOther curvature flowsScalar curvature

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 Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant (Xuezhang Chen, Pak Tung Ho and Liming Sun, 2016)8Chen 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)3Amacha 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)4Ho 2017 First eigenvalues of geometric operators under the Yamabe flow (Pak Tung Ho, 2018)6Ho 2018 Convergence rate of the weighted Yamabe flow (Pak Tung Ho, Jinwoo Shin and Zetian Yan, 2022)1Ho 2022 Convergence rate of the Q-curvature flow (Pak Tung Ho and Sanghoon Lee, 2024)2Ho 2024 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)7An-Lacroix 2025 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)5Afeltra 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence rate of the weighted Yamabe flow

    Pak Tung Ho, Jinwoo Shin and Zetian YanWider flows

    Topics Yamabe flow · Same authors Pak Tung Ho, Jinwoo Shin · Shared terms “convergence rate”, “Carlotto”, “Rubinstein”

  2. Convergence rate of the -curvature flow

    Pak Tung Ho and Sanghoon LeeWider flows

    Topics Yamabe flow, Other curvature flows · Same author Pak Tung Ho · Shared terms “convergence rate”, “Carlotto”, “Rubinstein”

  3. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “prescribed scalar”

  4. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow · Same author Pak Tung Ho

  5. A non-convergence phenomenon for the CR Yamabe flow

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow · Same author Pak Tung Ho

  6. First eigenvalues of geometric operators under the Yamabe flow

    Pak Tung HoWider flows

    Topics Yamabe flow, Scalar curvature · Same author Pak Tung Ho

  7. Convergence of the fractional Yamabe flow for arbitrary initial energy

    Jingeon An-Lacroix, Hardy Chan and Pak Tung HoWider flows

    Topics Yamabe flow · Same author Pak Tung Ho · Shared terms “inspired”

  8. Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

    Xuezhang Chen, Pak Tung Ho and Liming SunWider flows

    Topics Scalar curvature · Same author Pak Tung Ho · Shared terms “prescribed scalar”

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) 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) 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)

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.