Paper · Sep 16, 2023

Combinatorial curvature flows for generalized hyperbolic circle packings

Te Ba, Chao Zheng

math.DGWider flowsarXiv:2309.08901
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Abstract

Generalized circle packings were introduced in as a generalization of tangential circle packings in hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow, fractional combinatorial Calabi flow and combinatorial -th Calabi flow for generalized hyperbolic circle packings. We establish several equivalent conditions regarding the longtime behaviors of these flows. This provides effective algorithms for finding the generalized circle packings with prescribed total geodesic curvatures.

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

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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 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)1Xu 2022 Combinatorial Calabi flows for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)8Lei 2023 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)4Xu 2023 Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary (Xu Xu and Chao Zheng, 2023)5Xu 2023 Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary (Guangming Hu, Yi Qi, Yu Sun et al., 2023)7Hu 2023 Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry (Guangming Hu, Ziping Lei, Yi Qi et al., 2024)2Hu 2024 Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings (Guangming Hu, Ziping Lei, Yanlin Li et al., 2024)3Hu 2024 The Calabi flow with prescribed curvature on finite graphs (Yi Li, Jie Wang, Pingsan Yuan et al., 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 curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “circle packings”, “generalized circle”, “combinatorial curvature”

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

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

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Other curvature flows · Shared terms “circle packings”, “generalized circle”, “total geodesic”

  3. Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings

    Guangming Hu, Ziping Lei, Yanlin Li et al.Wider flows PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “circle packings”, “generalized hyperbolic”, “hyperbolic circle”

  4. 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, Long-time existence & convergence · Same author Chao Zheng · Shared terms “circle packings”, “combinatorial curvature”, “Calabi flow”

  5. Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “circle packings”, “combinatorial”, “generalized circle”

  6. The Calabi flow with prescribed curvature on finite graphs

    Yi Li, Jie Wang, Pingsan Yuan et al.Wider flows PDF

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

  7. Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

    Guangming Hu, Yi Qi, Yu Sun et al. PDF

    Topics Discrete & graph Ricci · Shared terms “hyperbolic circle”, “circle packings”, “generalized hyperbolic”

  8. 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 “combinatorial Calabi”, “Calabi flow”, “background geometry”

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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)

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.