Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Haozhao Li; Bing Wang; Kai Zheng

Paper · Nov 1, 2026

Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Haozhao Li, Bing Wang, Kai Zheng

math.DGWider flowsarXiv:2611.01882

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Abstract

We study global Calabi flows $g (t)$ on compact Kähler surfaces with a uniform Sobolev constant. For any sequence $t_{j} \to \infty$ , set $g_{j} (t) := g (t_{j} + t)$ . After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime $C^^{\infty}$ -Cheeger–Gromov topology: it is uniform in the Gromov–Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.

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

Other curvature flows Singularities Kähler geometry Limit spaces & RCD Kähler–Einstein & K-stability

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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) 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.

Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

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