Paper · Jun 8, 2025

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

Xiaorui Yang, Hao Yu

math.GTWider flowsarXiv:2506.07130v2
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Abstract

This paper presents a comprehensive study of the combinatorial -th Calabi flow for both finite and infinite ideal circle patterns. In the finite case, we establish a sharp criterion: the combinatorial -th Calabi flow with converges if and only if a constant curvature metric exists in the underlying geometric background. In the infinite setting, we prove the long-time existence of solutions to the combinatorial -th Calabi flow for , representing a significant advance in the theory of curvature flows on infinite structures.

Comments: Requesting withdrawal as co-authors need to refine the work before public release

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

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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 Calabi flows for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)2Lei 2023 Combinatorial Calabi flow for ideal circle pattern (Shengyu Li and Zhigang Wang, 2025)1Li 2025 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)3Zhang 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)4Ge 2025 Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns (Aijin Lin and Longxiang Wu, 2025)5Lin 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)8Li 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)7He 2025 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Combinatorial Calabi flow for ideal circle pattern

    Shengyu Li and Zhigang WangWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows, Kähler geometry · Shared terms “ideal circle”, “circle patterns”, “Calabi flow”

  2. Combinatorial Calabi flows for ideal circle patterns in spherical background geometry

    Ziping Lei and Puchun ZhouWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows, Kähler geometry · Shared terms “ideal circle”, “circle patterns”, “Calabi flow”

  3. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang PDF

    Topics Long-time existence & convergence, Other curvature flows, Kähler geometry · Shared terms “ideal circle”, “circle patterns”, “Calabi flow”

  4. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al. PDF

    Same author Hao Yu · Shared terms “infinite ideal”, “circle patterns”, “ideal circle”

  5. Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

    Aijin Lin and Longxiang WuWider flows PDF

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

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

    Haozhao Li, Bing Wang and Kai ZhengWider flows PDF

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

  7. 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 · Shared terms “Calabi flow”

  8. 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”

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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) 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) 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)