Paper · Jul 31, 2017

Stability of ALE Ricci-flat manifolds under Ricci flow

Alix Deruelle, Klaus Kroencke

math.DGarXiv:1707.09919v2
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Abstract

We prove that if an ALE Ricci-flat manifold is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to . By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

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Submitted
Jul 31, 2017
Last revised
Feb 28, 2020 (v2)
Subjects
math.DG, math.AP
DOI
10.1007/s12220-020-00376-4

Comments: 35 pages, final version, to appear in J. Geom. Anal

Ricci flowRicci-flat & Calabi–YauKähler geometryStability & uniquenessLong-time existence & convergence

Its neighbourhood

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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 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 Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones (Ronan J. Conlon and Alix Deruelle, 2020)8Conlon 2020 Long-time estimates for heat flows on ALE manifolds (Klaus Kroencke and Oliver Lindblad Petersen, 2020)6Kroencke 2020 A Łojasiewicz inequality for ALE metrics (Alix Deruelle and Tristan Ozuch, 2020)3Deruelle 2020 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)2Kroencke 2020 Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow (Alix Deruelle and Tristan Ozuch, 2021)1Deruelle 2021 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)5Deruelle 2024 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)7Kim 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

    Alix Deruelle and Tristan Ozuch

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

  2. 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 · Same author Klaus Kroencke · Shared terms “ALE”, “integrable”

  3. A Łojasiewicz inequality for ALE metrics

    Alix Deruelle and Tristan Ozuch

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

  4. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “ALE Ricci”

  5. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau, Long-time existence & convergence, Stability & uniqueness · Same author Alix Deruelle · Shared terms “ALE”, “linearly”

  6. Long-time estimates for heat flows on ALE manifolds

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Klaus Kroencke · Shared terms “ALE”

  7. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch

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

  8. Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

    Ronan J. Conlon and Alix Deruelle

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Alix Deruelle

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.

Foundational paper in the Ricci-flat & Calabi–Yau guide: Deruelle–Kröncke: stability of ALE Ricci-flat manifolds under Ricci flow.

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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… 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.