Paper · Sep 8, 2019

Circle patterns on surfaces of finite topological type

Huabin Ge, Bobo Hua, Ze Zhou

math.GTarXiv:1909.03419
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Abstract

This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.

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Submitted
Sep 8, 2019
Subjects
math.GT

Comments: 32 pages, 2 figures

Ricci flowSurfacesLong-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 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 On a combinatorial curvature for surfaces with inversive distance circle packing metrics (Huabin Ge and Xu Xu, 2017)8Ge 2017 Convergence of combinatorial Ricci flows to degenerate circle patterns (Asuka Takatsu, 2018)4Takatsu 2018 Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)5Feng 2020 Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)7Feng 2020 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)1Ge 2023 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)2Ge 2025 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)6Xiong 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)3Ge 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “circle patterns”, “combinatorial”, “Chow”

  2. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Surfaces · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”

  3. 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 · Same authors Huabin Ge, Bobo Hua · Shared terms “circle patterns”, “combinatorial Ricci”

  4. Convergence of combinatorial Ricci flows to degenerate circle patterns

    Asuka Takatsu

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

  5. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”, “Luo”

  6. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “circle patterns”, “intersection angles”, “Luo”

  7. Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”

  8. On a combinatorial curvature for surfaces with inversive distance circle packing metrics

    Huabin Ge and Xu Xu

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same author Huabin Ge · Shared terms “combinatorial Ricci”, “time behavior”, “equivalent”

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) 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) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

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.