Paper · May 21, 2026

Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

Haoyuan Sun

math.DGarXiv:2605.22347
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Abstract

We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.

Comments: 28 pages, comments welcome!

Ricci flowGeneralized & G₂ flows(Non)collapsingKähler geometryLimit spaces & RCDMinimal & CMC surfacesAnalytic 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 The Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)4Tosatti 2021 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)6Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)2Dang 2023 Hermitian null loci (Quang-Tuan Dang, 2024)3Dang 2024 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)7Wondo 2024 Long-time behavior of Ricci flow on some complex surfaces (John Lott, 2025)8Lott 2025 Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type (Haoyuan Sun, 2026)1Sun 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)5Angella 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    Haoyuan Sun PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, (Non)collapsing · Same author Haoyuan Sun · Shared terms “Chern Ricci”, “Green”, “local Kahler”

  2. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “Tosatti Weinkove”, “Hermitian”

  3. Hermitian null loci

    Quang-Tuan Dang PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, (Non)collapsing · Shared terms “null locus”, “Tosatti”, “Chern”

  4. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove PDF

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”

  5. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa PDF

    Topics Generalized & G₂ flows, Limit spaces & RCD · Shared terms “Chern Ricci”, “Green”, “Gromov Hausdorff”

  6. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “Hermitian”

  7. Singularity Types for Long-Time Chern-Ricci Flow

    Hosea Wondo PDF

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”

  8. Long-time behavior of Ricci flow on some complex surfaces

    John Lott PDF

    Topics Analytic minimal model program, Minimal & CMC surfaces

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.