Paper · Nov 17, 2023

Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

Guangming Hu, Yi Qi, Yu Sun, Puchun Zhou

math.GTarXiv:2311.10528
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Abstract

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the -skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdière's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

Comments: 26 pages, 7 figures

Ricci flowSingularitiesSurfacesStability & uniquenessDiscrete & graph Ricci

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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 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)8Xu 2022 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)7Ba 2023 Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry (Guangming Hu, Ziping Lei, Yi Qi et al., 2024)1Hu 2024 The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry (Guangming Hu, Ziping Lei, Yu Sun et al., 2024)3Hu 2024 Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry (Guangming Hu, Sicheng Lu, Dong Tan 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)5Hu 2024 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)6Ge 2025 A prescribed curvature flow on hyperbolic surfaces with infinite topological type (Xinrong Zhao and Puchun Zhou, 2025)4Zhao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Surfaces, Singularities · Same authors Guangming Hu, Yi Qi, Puchun Zhou · Shared terms “total geodesic”, “circle packings”, “geodesic curvature”

  2. Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry

    Guangming Hu, Sicheng Lu, Dong Tan et al. PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Guangming Hu, Puchun Zhou · Shared terms “circle packings”, “total geodesic”, “geodesic curvature”

  3. The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry

    Guangming Hu, Ziping Lei, Yu Sun et al. PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Guangming Hu, Yu Sun, Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “prescribed total”

  4. A prescribed curvature flow on hyperbolic surfaces with infinite topological type

    Xinrong Zhao and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Surfaces · Same author Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “cellular decomposition”

  5. 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, Stability & uniqueness · Same author Guangming Hu · Shared terms “circle packings”, “hyperbolic circle”, “generalized hyperbolic”

  6. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Surfaces, Stability & uniqueness · Same author Puchun Zhou · Shared terms “circle packings”

  7. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

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

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

    Xu Xu and Chao Zheng PDF

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

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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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.