Paper · Jan 3, 2025

Combinatorial Calabi flow for ideal circle pattern

Shengyu Li, Zhigang Wang

math.DGWider flowsarXiv:2501.01678v2Journal of Geometric Analysis, 2025
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Abstract

We study the combinatorial Calabi flow for ideal circle patterns in both hyperbolic and Euclidean background geometry. We prove that the flow exists for all time and converges exponentially fast to an ideal circle pattern metric on surfaces with prescribed attainable curvatures. As a consequence, we provide an algorithm to find the desired ideal circle patterns.

Comments: 14 pages,2 figures

Other curvature flowsSurfacesKähler geometryLong-time existence & convergence

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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 on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)8Luo 2021 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)5Ge 2023 Combinatorial Calabi flows for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)1Lei 2023 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)2Zhang 2025 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)4Xiong 2025 Prescribing hyperbolic bordered surfaces via combinatorial flows (Shengyu Li and Zhi-Gang Wang, 2025)7Li 2025 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)3Yang 2025 Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns (Aijin Lin and Longxiang Wu, 2025)6Lin 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”, “combinatorial Calabi”

  2. 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”, “combinatorial Calabi”

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

    Xiaorui Yang and Hao YuWider flows PDF

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

  4. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu PDF

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

  5. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

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

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

    Aijin Lin and Longxiang WuWider flows PDF

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

  7. Prescribing hyperbolic bordered surfaces via combinatorial flows

    Shengyu Li and Zhi-Gang WangWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows, Kähler geometry · Same author Shengyu Li · Shared terms “combinatorial”, “Calabi flow”, “converges exponentially”

  8. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Surfaces, Long-time existence & convergence, Other curvature flows · Shared terms “combinatorial Calabi”, “Calabi flow”, “algorithm”

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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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.