Calabi flow on polarized Kähler surfaces

Haozhao Li; Bing Wang; Kai Zheng

Paper · Nov 1, 2026

Calabi flow on polarized Kähler surfaces

Haozhao Li, Bing Wang, Kai Zheng

math.DGWider flowsarXiv:2611.01968

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Abstract

We study global smooth Calabi flows on polarized Kähler surfaces $(M, L)$ , assuming uniformly bounded Sobolev constants and the algebraic conditions $χ (O_{M}) = 1$ and $c_{1} (M) \cdot c_{1} (L) > 0$ . Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and $∣ S ∣ \leq 1$ . We illustrate the hypotheses with nonstationary flows near a product cscK metric on $P^{1} \times P^{1}$ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.

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Submitted: Nov 1, 2026
Subjects: math.DG, math.CV

Other curvature flows Kähler geometry Scalar curvature Limit spaces & RCD Stability & uniqueness Kähler–Einstein & K-stability

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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) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.
11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

[1] 6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

[2] 9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

[3] 11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

Calabi flow on polarized Kähler surfaces

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