Paper · Apr 3, 2026

The Calabi flow with prescribed curvature on finite graphs

Yi Li, Jie Wang, Pingsan Yuan, Chao Zheng

math.DGWider flowsarXiv:2604.02632
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Abstract

In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.

Other curvature flowsKähler geometryKähler–Einstein & K-stabilityDiscrete & graph Ricci

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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 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)5Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)1Ba 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)7Li 2023 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)6Yang 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)3Li 2025 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)2Lin 2026 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)8Cheng 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)4Hua 2026
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    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Other curvature flows · Same author Chao Zheng · Shared terms “Calabi flow”, “equivalent”

  2. The Ricci flow with prescribed curvature on graphs

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    Topics Discrete & graph Ricci, Other curvature flows · Shared terms “prescribed curvature”, “finite graph”, “Yau curvature”

  3. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows PDF

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

  4. The Ollivier Ricci flow with prescribed curvature on infinite graphs

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    Topics Discrete & graph Ricci · Shared terms “prescribed curvature”, “Lin”, “Yau”

  5. Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Other curvature flows · Same author Chao Zheng · Shared terms “Calabi flow”

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

    Xiaorui Yang and Hao YuWider flows PDF

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “constant curvature”

  7. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows PDF

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

  8. An improvement of regularity result for pseudo Calabi flow

    Jingrui Cheng and Junhao TianWider flows PDF

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “flow exists”

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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) 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) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.