Paper · Jun 17, 2021

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

Seunghyeok Kim, Monica Musso

math.APWider flowsarXiv:2106.09220v2
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Abstract

Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension is known to exist for all time and converges to a solution to the Yamabe problem as . We prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the Yamabe flow on any given Riemannian manifold of dimension , the resulting flow may blow up at multiple points on in the infinite time. Our proof is constructive, and indeed we construct such a flow by using solutions of the Yamabe problem on the unit sphere as blow-up profiles. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

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Submitted
Jun 17, 2021
Last revised
Jul 3, 2021 (v2)
Subjects
math.AP, math.DG

Comments: 54 pages, minor revision

Yamabe flowSingularitiesStability & uniquenessLong-time existence & convergencePhysics

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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)8Bahuaud 2016 Yamabe Flow with prescribed scalar curvature (Inas Amacha and Rachid Regbaoui, 2017)7Amacha 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)3Ho 2017 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)2Takahashi 2021 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)4An-Lacroix 2025 Symmetry Breaking and Moduli for Ancient Yamabe Flows on the Sphere (Haixia Chen, Seunghyeok Kim and Monica Musso, 2026)1Chen 2026 On the Yamabe flow in a bounded domain (Fei Fang and Zhong Tan, 2026)5Fang 2026 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. Symmetry Breaking and Moduli for Ancient Yamabe Flows on the Sphere

    Haixia Chen, Seunghyeok Kim and Monica MussoWider flows

    Topics Yamabe flow, Singularities · Same authors Seunghyeok Kim, Monica Musso · Shared terms “resulting flow”, “arbitrarily”

  2. 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”

  3. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

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

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

    Topics Yamabe flow, Physics · Shared terms “positive mass”

  5. On the Yamabe flow in a bounded domain

    Fei Fang and Zhong TanWider flows

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

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

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow · Shared terms “negativity”

  7. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

    Topics Yamabe flow, Long-time existence & convergence · Shared terms “compact Riemannian”, “Riemannian manifold”

  8. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

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

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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.