Paper · Oct 4, 2026

Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Haozhao Li, Bing Wang, Kai Zheng

math.DGWider flowsarXiv:2610.05217
PDF Abstract page

As on arXiv

Abstract

We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.

Comments: 34 pages

Other curvature flowsSingularitiesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability

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 Hermitian Calabi functional in complexified orbits (Jie He and Kai Zheng, 2023)5He 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)1Li 2023 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)8Yang 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)2Li 2025 Twisted Calabi functional and twisted Calabi flow (Jie He and Haozhao Li, 2025)6He 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)3He 2025 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)7Cheng 2026 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)4Tian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same authors Haozhao Li, Kai Zheng · Shared terms “Calabi flow”, “uniformly bounded”

  2. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows PDF

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Haozhao Li · Shared terms “Calabi flow”, “compact Kähler”

  3. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

    Jie He and Haozhao LiWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Haozhao Li · Shared terms “Calabi flow”, “compact Kähler”

  4. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Kai Zheng · Shared terms “Calabi flow”

  5. Hermitian Calabi functional in complexified orbits

    Jie He and Kai ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Kai Zheng · Shared terms “Calabi”

  6. Twisted Calabi functional and twisted Calabi flow

    Jie He and Haozhao LiWider flows PDF

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Haozhao Li · Shared terms “Calabi flow”, “compact Kähler”, “time existence”

  7. An improvement of regularity result for pseudo Calabi flow

    Jingrui Cheng and Junhao TianWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Calabi flow”

  8. Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

    Xiaorui Yang and Hao YuWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Calabi flow”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 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.