Paper · Mar 8, 2022

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

Charles Cifarelli, Ronan J. Conlon, Alix Deruelle

math.DGarXiv:2203.04380v3
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Abstract

We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient Kähler-Ricci soliton with soliton metric with bounded scalar curvature whose soliton vector field has an integral curve along which is biholomorphic to either or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

Comments: 32 pages; oversight in the proof of Theorem B rectified (uniform lower bound on volume of curves)

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularitiesKähler geometryScalar curvatureBoundaries & non-compact

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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 An Aubin continuity path for shrinking gradient Kähler-Ricci solitons (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)8Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)2Bamler 2022 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)6Conlon 2025 An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for t=0 (Ivin Babu, Ronan J. Conlon and Alix Deruelle, 2025)5Babu 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)4Conlon 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 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 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)3Conlon 2026
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  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, Kähler geometry, Singularities · Same authors Charles Cifarelli, Ronan J. Conlon · Shared terms “Kähler surfaces”, “compact Kähler”, “finite time”

  2. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al. PDF

    Topics Kähler–Ricci solitons, Scalar curvature, Kähler geometry · Same authors Charles Cifarelli, Ronan J. Conlon, Alix Deruelle · Shared terms “dimensional shrinking”, “shrinking gradient”, “gradient Kähler”

  3. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “soliton vector”, “shrinking gradient”, “gradient Kähler”

  4. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”

  5. An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for

    Ivin Babu, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”

  6. 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 Kähler geometry, Singularities, Ricci solitons · Same author Ronan J. Conlon · Shared terms “Kähler surfaces”, “compact Kähler”, “finite time”

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

    Tongxin Xu and Zhenlei Zhang PDF

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

  8. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Singularities · Same authors Charles Cifarelli, Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient 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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

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.