Paper · Mar 15, 2025

Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

Sigurd B. Angenent, Dan Knopf

math.DGarXiv:2503.12210
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Abstract

Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have uncountably many unstable perturbations.

Ricci flowRicci-flat & Calabi–YauStability & uniquenessBoundaries & non-compact

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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 On a dichotomy of the curvature decay of steady Ricci soliton (Pak-Yeung Chan and Bo Zhu, 2021)8Chan 2021 Robinson-Trautman solutions with scalar hair and Ricci flow (Masato Nozawa and Takashi Torii, 2023)6Nozawa 2023 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)7Kroencke 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)1Garfinkle 2024 L^2-instability of the Taub-Bolt metric under the Ricci flow (John Hughes, 2024)2Hughes 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)5Isenberg 2025 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)4Kim 2025 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al. PDF

    Topics Stability & uniqueness · Same author Dan Knopf · Shared terms “unstable perturbations”, “stationary”

  2. L^2-instability of the Taub-Bolt metric under the Ricci flow

    John Hughes PDF

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact · Shared terms “instability”, “perturbation”

  3. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “stationary”

  4. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “unstable”

  5. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics Boundaries & non-compact · Same author Dan Knopf

  6. Robinson-Trautman solutions with scalar hair and Ricci flow

    Masato Nozawa and Takashi Torii PDF

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact · Shared terms “dynamical”

  7. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz PDF

    Topics Boundaries & non-compact, Stability & uniqueness · Shared terms “instability”, “dynamical”

  8. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu PDF

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) 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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.