Paper · Dec 3, 2024

Generalized Yamabe Flows

Jørgen Olsen Lye, Boris Vertman, Mannaim Gennaro Vitti

math.DGWider flowsarXiv:2412.02593v2
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Abstract

In this work we introduce a family of conformal flows generalizing the classical Yamabe flow. We prove that for a large class of such flows long-time existence holds, and the arguments are in fact simpler than in the classical case. Moreover, we establish convergence for the case of negative scalar curvature and expect a similar statement for the positive and the flat cases as well.

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Submitted
Dec 3, 2024
Last revised
Aug 22, 2025 (v2)
Subjects
math.DG
DOI
10.1007/s13324-025-01121-2

Comments: V2: Added some clarification about the potential difference between the background metric and the initial metric. Corrected typos

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 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)2Takahashi 2021 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)1Yan 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)6Li 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)7Carron 2022 Convergence rate of the weighted Yamabe flow (Pak Tung Ho, Jinwoo Shin and Zetian Yan, 2022)8Ho 2022 Existence of the (α,β)-Ricci-Yamabe flow on closed manifolds (Liangdi Zhang, 2023)5Zhang 2023 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)4Afeltra 2026 A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree (Weike Yu, 2026)3Yu 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

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

  2. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Long-time existence & convergence

  3. A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree

    Weike YuWider flows

    Topics Yamabe flow, Scalar curvature

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow

  5. Existence of the -Ricci-Yamabe flow on closed manifolds

    Liangdi Zhang

    Topics Yamabe flow, Long-time existence & convergence

  6. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows

    Topics Yamabe flow, Long-time existence & convergence

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

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow

  8. Convergence rate of the weighted Yamabe flow

    Pak Tung Ho, Jinwoo Shin and Zetian YanWider 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 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) 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.