Paper · Aug 7, 2022

Combinatorial curvature flows for generalized circle packings on surfaces with boundary

Xu Xu, Chao Zheng

math.DGarXiv:2208.03643v2
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Abstract

In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the type generalized circle packing metric introduced by Guo-Luo. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

Comments: To appear in IMRN

Ricci flowOther curvature flowsSurfacesKähler geometryLong-time existence & convergenceDiscrete & 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 Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)5Luo 2021 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)8Xu 2021 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)1Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)3Ba 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)7Xu 2023 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)6Xiong 2025 Discrete conformal structures on surfaces with boundary (III) – Deformation (Xu Xu and Chao Zheng, 2025)2Xu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

    Xu Xu and Chao Zheng PDF

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

  2. Discrete conformal structures on surfaces with boundary (III) – Deformation

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Same author Chao Zheng · Shared terms “prescribed lengths”, “combinatorial curvature”, “geodesic boundaries”

  3. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

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

  4. Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

    Xu Xu and Chao Zheng PDF

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

  5. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “geodesic boundaries”, “combinatorial”, “ideally triangulated”

  6. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “generalized circle”, “Guo Luo”, “combinatorial curvature”

  7. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

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

  8. A new class of discrete conformal structures on surfaces with boundary

    Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Other curvature flows · Shared terms “combinatorial curvature”, “prescribed lengths”, “geodesic boundaries”

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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.

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) 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)

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. 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.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.