Paper · Sep 17, 2025

An -Regularity Theorem for Non-collapsed Ricci Flow

Harry Fluck, Max Hallgren

math.DGarXiv:2509.14154v2
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Abstract

In this article we prove an -regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano Kähler-Ricci flows. In the course of our proof, we find a criterion for uniform convergence of solutions to the heat equation along a sequence of -converging Ricci flows, and apply this to new parabolic regularizations of some natural geometric quantities.

Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry

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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 Tangent Flows of Kähler Metric Flows (Max Hallgren and Wangjian Jian, 2022)8Hallgren 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)3Hallgren 2023 Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Ronan J. Conlon, Max Hallgren and Zilu Ma, 2025)2Conlon 2025 Remarks on Singular Kähler-Einstein Metrics (Max Hallgren and Gábor Székelyhidi, 2025)6Hallgren 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)5Hallgren 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)7Chen 2026 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)1Cifarelli 2026 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)4Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al. PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren

  2. Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Ronan J. Conlon, Max Hallgren and Zilu Ma PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “collapsed”

  3. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “sequence”

  4. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “collapsed”

  5. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren · Shared terms “Fano”

  6. Remarks on Singular Kähler-Einstein Metrics

    Max Hallgren and Gábor Székelyhidi PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren · Shared terms “collapsed”, “sequence”

  7. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren

  8. Tangent Flows of Kähler Metric Flows

    Max Hallgren and Wangjian Jian PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “parabolic regularizations”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.