Paper · Apr 5, 2026

Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

Longteng Chen, Max Hallgren, Lucas Lavoyer

math.DGarXiv:2604.04223v3
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Abstract

Let be a compact analytic space with a finite number of singular points, where the metric at each singular point is modelled on a Kähler cone with smooth canonical model. We show that the Kähler-Ricci flow with such initial data satisfies a curvature bound, and that the flow near each singular point is modelled on the unique Kähler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the Kähler–Ricci flow emerging from singularities arising in the analytic minimal model program.

Comments: Streamlined proofs and minor corrections. 35 pages, 2 figures. Comments are welcome!

Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryAnalytic minimal model program

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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 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)8Hallgren 2023 Uniqueness of asymptotically conical Kähler-Ricci flow (Longteng Chen, 2025)6Chen 2025 Stability of asymptotically conical gradient Kähler-Ricci expanders (Longteng Chen, 2025)5Chen 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)3Hallgren 2026 Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow (Longteng Chen, 2026)4Chen 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)1Chen 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)7Cifarelli 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)2Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same authors Longteng Chen, Max Hallgren, Lucas Lavoyer · Shared terms “conical singularities”, “expander”

  2. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry

  3. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Max Hallgren

  4. Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow

    Longteng Chen PDF

    Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Longteng Chen · Shared terms “Ricci expander”, “initial data”

  5. Stability of asymptotically conical gradient Kähler-Ricci expanders

    Longteng Chen PDF

    Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Longteng Chen · Shared terms “Ricci expander”

  6. Uniqueness of asymptotically conical Kähler-Ricci flow

    Longteng Chen PDF

    Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Longteng Chen · Shared terms “flow emerging”, “Kähler cone”

  7. 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, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “curvature bounds”

  8. 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

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) 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) 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)

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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.