Paper · Sep 15, 2018

Convergence of the fractional Yamabe flow for a class of initial data

Hardy Chan, Yannick Sire, Liming Sun

math.APWider flowsarXiv:1809.05753
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Abstract

This work is a follow-up on the work of the second author with P. Daskalopoulos and J.L. Vázquez. In this latter work, we introduced the Yamabe flow associated to the so-called fractional curvature and prove some existence result of mild (semi-group) solutions. In the present work, we continue this study by proving that for some class of data one can prove actually convergence of the flow in a more general context. We build on the approach simplified in the book of M. Struwe.

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Submitted
Sep 15, 2018
Subjects
math.AP
DOI
10.2422/2036-2145.201809_007
Yamabe flowAnisotropic & nonlocal flows

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)3Almaraz 2016 Existence and properties of ancient solutions of the Yamabe flow (Shu-Yu Hsu, 2016)8Hsu 2016 Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds (Panagiota Daskalopoulos, Yannick Sire and Juan-Luis Vázquez, 2017)1Daskalopoulo 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)4Ho 2017 Yamabe flow on modified Riemann extension (Harish D, 2019)5D 2019 Convergence rate of the weighted Yamabe flow (Pak Tung Ho, Jinwoo Shin and Zetian Yan, 2022)7Ho 2022 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)2An-Lacroix 2025 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)6Afeltra 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds

    Panagiota Daskalopoulos, Yannick Sire and Juan-Luis VázquezWider flows

    Topics Yamabe flow, Anisotropic & nonlocal flows · Same author Yannick Sire · Shared terms “fractional Yamabe”, “fractional curvature”

  2. 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 Hardy Chan · Shared terms “fractional Yamabe”

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

    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow · Same author Liming Sun

  4. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  5. Yamabe flow on modified Riemann extension

    Harish DWider flows

    Topics Yamabe flow

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow

  7. Convergence rate of the weighted Yamabe flow

    Pak Tung Ho, Jinwoo Shin and Zetian YanWider flows

    Topics Yamabe flow

  8. Existence and properties of ancient solutions of the Yamabe flow

    Shu-Yu HsuWider 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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978)

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.