Paper · Apr 21, 2021

Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

Alix Deruelle, Tristan Ozuch

math.DGarXiv:2104.10630
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Abstract

We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's -functional adapted to the ALE situation and denoted . This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.

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Submitted
Apr 21, 2021
Subjects
math.DG

Comments: 56 pages, comments welcome!

Ricci flowRicci-flat & Calabi–YauStability & uniqueness

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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 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 Stability of ALE Ricci-flat manifolds under Ricci flow (Alix Deruelle and Klaus Kroencke, 2017)3Deruelle 2017 A Łojasiewicz inequality for ALE metrics (Alix Deruelle and Tristan Ozuch, 2020)1Deruelle 2020 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)6Kroencke 2020 On the dynamical behaviour of the generalized Ricci flow (Alberto Raffero and Luigi Vezzoni, 2020)7Raffero 2020 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)4Lopez 2024 Orbifold singularity formation along ancient and immortal Ricci flows (Alix Deruelle and Tristan Ozuch, 2024)2Deruelle 2024 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)5Kim 2025 Integrable Deformations and Stability of the Ricci Flow (Maxwell Stolarski and Alex Waldron, 2026)8Stolarski 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Łojasiewicz inequality for ALE metrics

    Alix Deruelle and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau · Same authors Alix Deruelle, Tristan Ozuch · Shared terms “ALE metrics”, “flat ALE”, “Lojasiewicz”

  2. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same authors Alix Deruelle, Tristan Ozuch · Shared terms “ALE metrics”, “flat ALE”, “along Ricci”

  3. Stability of ALE Ricci-flat manifolds under Ricci flow

    Alix Deruelle and Klaus Kroencke

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Alix Deruelle · Shared terms “ALE Ricci”, “integrable”

  4. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau · Same author Tristan Ozuch · Shared terms “ALE Ricci”, “ALE metrics”, “Perelman”

  5. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Tristan Ozuch · Shared terms “along Ricci”, “Perelman”

  6. Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

    Klaus Kroencke and Oliver Lindblad Petersen

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

  7. On the dynamical behaviour of the generalized Ricci flow

    Alberto Raffero and Luigi Vezzoni

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

  8. Integrable Deformations and Stability of the Ricci Flow

    Maxwell Stolarski and Alex Waldron

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

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)

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.