Paper · Oct 27, 2018

Combinatorial -th Calabi flows on surfaces

Aijin Lin, Xiaoxiao Zhang

math.DGWider flowsarXiv:1810.11625
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Abstract

For triangulated surfaces and any , we introduce the combinatorial -th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when . The difficulties for the generalizations come from the nonlinearity of the -th flow equation when . Adopting different approaches, we show that the solution to the combinatorial -th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H. Ge, Ge-Xu and Ge-Hua on the combinatorial Calabi flow from to any .

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Submitted
Oct 27, 2018
Subjects
math.DG

Comments: 24 pages

Other curvature flowsSurfacesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stabilityDiscrete & graph Ricci

Its neighbourhood

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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 2016, 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)4Luo 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)7Xu 2022 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)3Ba 2023 Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry (Guangming Hu, Ziping Lei, Yi Qi et al., 2024)5Hu 2024 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)1Zhang 2025 Combinatorial Calabi flow for ideal circle pattern (Shengyu Li and Zhigang Wang, 2025)6Li 2025 Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns (Aijin Lin and Longxiang Wu, 2025)2Lin 2025 Combinatorial Calabi flows on infinitely triangulated surfaces (Yuerong Bian and Xu Xu, 2026)8Bian 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 flows with ideal circle patterns

    Xiaoxiao Zhang

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Same author Xiaoxiao Zhang · Shared terms “Calabi flow”, “combinatorial Calabi”, “hyperbolic resp”

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

    Aijin Lin and Longxiang WuWider flows

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Aijin Lin · Shared terms “Calabi flow”, “combinatorial Calabi”, “background geometry”

  3. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Long-time existence & convergence · Shared terms “Calabi flow”, “combinatorial”, “circle packings”

  4. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “Calabi flow”, “combinatorial Calabi”, “triangulated surfaces”

  5. Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry

    Guangming Hu, Ziping Lei, Yi Qi et al.Wider flows

    Topics Discrete & graph Ricci, Surfaces, Kähler–Einstein & K-stability · Shared terms “Calabi flow”, “combinatorial”, “circle packings”

  6. Combinatorial Calabi flow for ideal circle pattern

    Shengyu Li and Zhigang WangWider flows

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

  7. Combinatorial curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “combinatorial”, “circle packings”, “triangulated surfaces”

  8. Combinatorial Calabi flows on infinitely triangulated surfaces

    Yuerong Bian and Xu XuWider flows

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “combinatorial Calabi”, “triangulated surfaces”

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.

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 whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… 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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.