Paper · Oct 19, 2022

The modified Kähler-Ricci flow, II

Haotian Wu, Zhou Zhang

math.DGarXiv:2210.10296
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Abstract

We improve the understanding of both finite time and infinite time singularities of the modified Kähler-Ricci flow as initiated by the second author of this paper in [26]. This is done by relating the modified Kähler-Ricci flow with the recent studies on the classic Kähler-Ricci flow and the degenerate complex Monge-Ampère equation.

Comments: 20 pages. Comments welcome

Ricci flowKähler–Ricci flowSingularitiesKähler geometry

Its neighbourhood

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 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)1Zhao 2023 Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows (Hosea Wondo and Zhou Zhang, 2023)8Wondo 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)5Cho 2024 Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities (Alexander Bednarek, 2024)3Bednarek 2024 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)2George 2024 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)7Zhang 2025 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)6Cifarelli 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. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  2. Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds

    Mathew George PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  3. Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

    Alexander Bednarek PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  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, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  5. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  6. 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 · Shared terms “time singularity”, “finite time”

  7. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  8. Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

    Hosea Wondo and Zhou Zhang PDF

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

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) 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. 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.