Paper · Jan 3, 2025

Combinatorial Calabi flows with ideal circle patterns

Xiaoxiao Zhang

math.DGarXiv:2501.01605
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Abstract

In this paper, we extend the work of Ge-Hua-Zhou on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the combinatorial Calabi flows with any given initial Euclidean (hyperbolic resp.)ideal circle pattern exists for all time and converges exponentially fast to a flat cone metric (hyperbolic resp.) on a given surface.

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Submitted
Jan 3, 2025
Subjects
math.DG, math.GT
Ricci flowOther curvature flowsKähler geometryLong-time existence & convergenceDiscrete & graph Ricci

Its neighbourhood

Connected papers

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 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)3Ge 2023 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 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)5Xiong 2025 Infinite combinatorial Ricci flow in spherical background geometry (Chang Li, Yangxiang Lu and Hao Yu, 2025)8Li 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)7Ge 2025 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)4Yang 2025 The character of ideal circle patterns (Chang Li, Aijin Lin and Liangming Shen, 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

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

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

    Ziping Lei and Puchun ZhouWider flows

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

  3. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Shared terms “circle patterns”, “ideal circle”, “combinatorial”

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

    Xiaorui Yang and Hao YuWider flows

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

  5. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu

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

  6. The character of ideal circle patterns

    Chang Li, Aijin Lin and Liangming Shen

    Topics Discrete & graph Ricci · Shared terms “ideal circle”, “circle patterns”, “Hua Zhou”

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

    Huabin Ge, Bobo Hua, Hao Yu et al.

    Topics Discrete & graph Ricci · Shared terms “circle patterns”, “ideal circle”, “combinatorial”

  8. Infinite combinatorial Ricci flow in spherical background geometry

    Chang Li, Yangxiang Lu and Hao Yu

    Topics Discrete & graph Ricci, Other curvature flows · Shared terms “circle patterns”, “combinatorial”, “background geometry”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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.

  2. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.