Paper · Sep 24, 2020

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

Klaus Kroencke, Oliver Lindblad Petersen

math.DGarXiv:2009.11854v2
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Abstract

We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in , for any , where is the dimension of the manifold. In particular, our result applies to all known examples of -dimensional gravitational instantons. Our decay rates are strong enough to prove positive scalar curvature rigidity in , for each , generalizing a result by Appleton.

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Submitted
Sep 24, 2020
Last revised
Jul 21, 2026 (v2)
Subjects
math.DG, math.AP
Journal
Adv. Math. 502 Part B (2026), 111139
DOI
10.1016/j.aim.2026.111139

Comments: 99 pages, published version

Ricci flowRicci-flat & Calabi–YauScalar curvatureStability & uniqueness

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 Stability of ALE Ricci-flat manifolds under Ricci flow (Alix Deruelle and Klaus Kroencke, 2017)2Deruelle 2017 Long-time estimates for heat flows on ALE manifolds (Klaus Kroencke and Oliver Lindblad Petersen, 2020)1Kroencke 2020 A Łojasiewicz inequality for ALE metrics (Alix Deruelle and Tristan Ozuch, 2020)5Deruelle 2020 A Survey on the Ricci flow on Singular Spaces (Klaus Kroencke and Boris Vertman, 2021)6Kroencke 2021 Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow (Alix Deruelle and Tristan Ozuch, 2021)4Deruelle 2021 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)7Jiang 2021 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)8Lopez 2024 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)3Kroencke 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time estimates for heat flows on ALE manifolds

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same authors Klaus Kroencke, Oliver Lindblad Petersen · Shared terms “ALE manifolds”, “parallel spinor”

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

    Alix Deruelle and Klaus Kroencke

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

  3. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Scalar curvature, Stability & uniqueness · Same author Klaus Kroencke · Shared terms “curvature rigidity”

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

    Alix Deruelle and Tristan Ozuch

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

  5. A Łojasiewicz inequality for ALE metrics

    Alix Deruelle and Tristan Ozuch

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

  6. A Survey on the Ricci flow on Singular Spaces

    Klaus Kroencke and Boris Vertman

    Topics Ricci-flat & Calabi–Yau, Scalar curvature · Same author Klaus Kroencke · Shared terms “positive scalar”

  7. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang

    Topics Ricci-flat & Calabi–Yau, Scalar curvature · Shared terms “initial metric”

  8. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch

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

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 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. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 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.