Paper · Sep 28, 2022

Convergence rate of the weighted Yamabe flow

Pak Tung Ho, Jinwoo Shin, Zetian Yan

math.DGWider flowsarXiv:2212.04367
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Abstract

The weighted Yamabe flow was the geometric flow introduced to study the weighted Yamabe problem on smooth metric measure spaces. 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 weighted Yamabe flow.

Comments: arXiv admin note: substantial text overlap with arXiv:2107.09616. substantial text overlap with arXiv:1401.3738 by other authors

Yamabe flowLimit spaces & RCD

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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 2021, 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 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)8Takahashi 2021 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)1Yan 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)6Carron 2022 A dual Yamabe flow and related integral flows (Jingang Xiong, 2023)5Xiong 2023 Convergence rate of the Q-curvature flow (Pak Tung Ho and Sanghoon Lee, 2024)2Ho 2024 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)7Lye 2024 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)4An-Lacroix 2025 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)3Afeltra 2026
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 Weighted Yamabe Flow

    Zetian YanWider flows PDF

    Topics Yamabe flow, Limit spaces & RCD · Same author Zetian Yan · Shared terms “weighted Yamabe”, “smooth metric”, “metric measure”

  2. Convergence rate of the -curvature flow

    Pak Tung Ho and Sanghoon LeeWider flows PDF

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

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows PDF

    Topics Yamabe flow · Same author Pak Tung Ho

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

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

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

  5. A dual Yamabe flow and related integral flows

    Jingang XiongWider flows PDF

    Topics Yamabe flow, Limit spaces & RCD

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

    Gilles Carron, Eric Chen and Yi WangWider flows PDF

    Topics Yamabe flow

  7. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows PDF

    Topics Yamabe flow

  8. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows PDF

    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 within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.