Paper · Sep 18, 2025

Laplace comparison on Kähler Ricci flow and convergence

Gang Tian, Qi S. Zhang, Zhenlei Zhang, Meng Zhu, Xiaohua Zhu

math.DGarXiv:2509.14820v2
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As on arXiv

Abstract

We first prove a uniform integral Laplace comparison result for the Kähler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.

Comments: 43 pages. replaces the appendix by a remark and link

Ricci flowKähler–Ricci flow4-manifoldsKähler geometry

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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 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)3Tian 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)2Tian 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)4Wang 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)5Hallgren 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)7Jian 2023 Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)" (Xiuxiong Chen and Bing Wang, 2025)1Chen 2025 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)6Ye 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)8Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"

    Xiuxiong Chen and Bing Wang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Laplace comparison”

  2. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano”, “initial metric”

  3. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano manifold”, “Tian”

  4. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”

  5. 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 · Same author Gang Tian · Shared terms “singular set”

  6. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “comparison”, “Tian”

  7. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Gang Tian · Shared terms “Fano manifold”

  8. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.