Paper · Nov 5, 2021

Infinite-time incompleteness of noncompact Yamabe flow

Jin Takahashi, Hikaru Yamamoto

math.DGWider flowsarXiv:2111.03222
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Abstract

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

Comments: 21 pages, 1 figure

Yamabe flowSingularitiesBoundaries & non-compactLong-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 Convergence of the Weighted Yamabe Flow (Zetian Yan, 2021)6Yan 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)8Li 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)4Carron 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)3Lye 2024 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)2Ji 2025 On the Yamabe flow in a bounded domain (Fei Fang and Zhong Tan, 2026)1Fang 2026 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)7Afeltra 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Yamabe flow in a bounded domain

    Fei Fang and Zhong TanWider flows PDF

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

  2. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence, Boundaries & non-compact

  3. Generalized Yamabe Flows

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

    Topics Yamabe flow, Long-time existence & convergence

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

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    Topics Yamabe flow, Boundaries & non-compact

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

    Liangdi Zhang PDF

    Topics Yamabe flow, Long-time existence & convergence

  6. Convergence of the Weighted Yamabe Flow

    Zetian YanWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows PDF

    Topics Yamabe flow

  8. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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. 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.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.