Paper · Aug 18, 2026

Finite Time Type I Singularities of the Kähler Ricci Flow

Tongxin Xu, Zhenlei Zhang

math.DGarXiv:2608.17458v2
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Abstract

We prove the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow. More precisely, for any compact Kähler manifold and its blow-up , if , then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker .

Comments: 18 pages; updated acknowledgments

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 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)3Cifarelli 2022 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)5Zhang 2026 Compactness and Rigidity of Complete Kähler Ricci Shrinkers (Tongxin Xu and Zhenlei Zhang, 2026)4Xu 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)2Xu 2026 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)7Jian 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)8Jian 2026 Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature (Tongxin Xu and Zhenlei Zhang, 2026)6Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Kähler geometry, Singularities · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “finite time”, “compact Kähler”

  2. Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “finite time”

  3. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time Type”, “finite time”, “Knopf conjecture”

  4. Compactness and Rigidity of Complete Kähler Ricci Shrinkers

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang

  5. Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

    Lei Zhang and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Zhenlei Zhang · Shared terms “compact Kähler”

  6. Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “compact Kähler”

  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 · Shared terms “finite time”, “compact Kähler”

  8. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “finite time”, “compact Kähler”

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.