Paper · Jul 26, 2024

Ancient and expanding spin ALE Ricci flows

Isaac M. Lopez, Tristan Ozuch

math.DGarXiv:2407.18438
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Abstract

We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in and mild decay at infinity are hyperkähler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's -functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized -functional related to a notion of weighted mass.

Comments: 21 pages

Ricci flowRicci solitonsAncient solutionsRicci-flat & Calabi–YauKähler geometryScalar curvature

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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 Spinors and mass on weighted manifolds (Julius Baldauf and Tristan Ozuch, 2022)4Baldauf 2022 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)6Leandro 2022 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)8Zhang 2024 Orbifold singularity formation along ancient and immortal Ricci flows (Alix Deruelle and Tristan Ozuch, 2024)1Deruelle 2024 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)2Kim 2025 Dynamical functionals on ancient ARF Ricci flows (Isaac M. Lopez and Rio Schillmoeller, 2025)3Lopez 2025 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)5Deng 2026 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II (Reto Buzano and Gianmichele Di Matteo, 2026)7Buzano 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions, Ricci solitons · Same author Tristan Ozuch · Shared terms “ALE”, “orbifold”

  2. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Tristan Ozuch · Shared terms “renormalized”, “hyperkähler”, “Perelman”

  3. Dynamical functionals on ancient ARF Ricci flows

    Isaac M. Lopez and Rio Schillmoeller PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions · Same author Isaac M. Lopez

  4. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch PDF

    Topics Scalar curvature · Same author Tristan Ozuch · Shared terms “weighted mass”, “Perelman”

  5. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions

  6. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions, Scalar curvature · Shared terms “controlled”

  7. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II

    Reto Buzano and Gianmichele Di Matteo PDF

    Topics Ancient solutions, Scalar curvature

  8. A rigidity result for ancient Ricci flows

    Qi S. Zhang PDF

    Topics Ancient solutions

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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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.