Paper · Apr 12, 2025

Generalized circle patterns on surfaces with cusps

Zhiwen Xiong, Xu Xu

math.GTarXiv:2504.09172
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Abstract

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

Comments: 18 pages, 2 figures

Ricci flowOther curvature flowsSurfacesKähler geometryStability & uniquenessLong-time existence & convergenceDiscrete & graph Ricci

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

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    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Shared terms “circle patterns”, “combinatorial Calabi”, “Calabi flow”

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    Xu Xu and Chao Zheng PDF

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

  4. Combinatorial Calabi flow for ideal circle pattern

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

  5. 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 · Shared terms “generalized circle”, “hyperbolic circle”, “generalized hyperbolic”

  6. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Shared terms “circle patterns”, “combinatorial curvature”, “Luo”

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

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    Topics Discrete & graph Ricci, Surfaces, Stability & uniqueness · Shared terms “hyperbolic circle”, “generalized circle”, “generalized hyperbolic”

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

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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.