Paper · Jun 22, 2026

On the Yamabe flow in a bounded domain

Fei Fang, Zhong Tan

math.APWider flowsarXiv:2606.22815
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Abstract

This paper investigates the dynamical behaviors of solutions to the Yamabe flow via the modified potential well method. We first establish the local existence and regularity of weak solutions for the flow. Several new results concerning global existence and blowup are obtained by classifying initial data into stable and unstable sets. Specifically, solutions with initial data in the stable set exist globally and extinguish in finite time, whereas those originating from unstable initial data blow up in infinite time. For certain high-energy initial data, we show that the solution decays to zero as time tends to infinity and undergoes finite-time blowup. In addition, we analyze Palais-Smale sequences to reveal the intrinsic relationship between the long-time asymptotic behavior of solutions and steady states. Finally, we derive the Pohozaev identity for the equation and prove the corresponding nonexistence theorem.

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Submitted
Jun 22, 2026
Subjects
math.AP
Yamabe flowSingularitiesWeak & singular flowsStability & uniquenessLong-time existence & convergence

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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)1Takahashi 2021 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)4Yan 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)3Li 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)8Carron 2022 Existence of the (α,β)-Ricci-Yamabe flow on closed manifolds (Liangdi Zhang, 2023)5Zhang 2023 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)2Lye 2024 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)7Ji 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. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Long-time existence & convergence, Singularities · Shared terms “infinite time”

  2. Generalized Yamabe Flows

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  3. Combinatorial Yamabe flow on hyperbolic bordered surfaces

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    Topics Yamabe flow, Long-time existence & convergence

  4. Convergence of the Weighted Yamabe Flow

    Zetian YanWider flows

    Topics Yamabe flow, Long-time existence & convergence

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

    Liangdi Zhang

    Topics Yamabe flow, Long-time existence & convergence

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow

  7. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows

    Topics Yamabe flow, Long-time existence & convergence, Stability & uniqueness

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

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    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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.