Paper · Aug 24, 2023

Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

Hosea Wondo, Zhou Zhang

math.DGarXiv:2308.12527v2Geom. Topol. 29 (2025) 479-494
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Abstract

In this paper, we show that the singularity type of solutions to the Käher-Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely if there exists a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises previous results by Y. Zhang for the semi-ample case.

Comments: 14 pages, 2 figures, all comments welcomed. Corrected typos and modified presentation of section 4

Ricci flowKähler–Ricci flowSingularitiesKähler geometryLong-time existence & convergenceAnalytic minimal model programNumerics & visualization

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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 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)2Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)4Fung 2021 Calabi Symmetry and the Continuity Method (Hosea Wondo, 2022)6Wondo 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)3Wu 2022 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)1Wondo 2024 Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)8Chen 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)5Xu 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)7Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity Types for Long-Time Chern-Ricci Flow

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Hosea Wondo · Shared terms “singularity type”, “independence”, “solution starting”

  2. Curvature Estimates for the Continuity Method

    Hosea Wondo PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Same author Hosea Wondo · Shared terms “semi ample”

  3. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Zhou Zhang

  4. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “semi ample”

  5. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang PDF

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

  6. Calabi Symmetry and the Continuity Method

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Hosea Wondo

  7. 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

  8. Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

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

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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) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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

  3. 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.