Paper · May 9, 2025

Infinite combinatorial Ricci flow in spherical background geometry

Chang Li, Yangxiang Lu, Hao Yu

math.GTarXiv:2505.05925v2
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Abstract

Since the fundamental work of Chow-Luo, Ge et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci flow with prescribed total geodesic curvatures in spherical background geometry. For infinite cellular decompositions, we establish the existence of a solution to the flow equation for all time. Furthermore, under an additional condition, we prove that the solution converges as time tends to infinity. To the best of our knowledge, this is the first study of an infinite combinatorial curvature flow in spherical background geometry.

Ricci flowOther curvature flowsDiscrete & 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 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)1Ge 2023 Combinatorial Calabi flows for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)8Lei 2023 The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry (Guangming Hu, Ziping Lei, Yu Sun et al., 2024)2Hu 2024 Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry (Guangming Hu, Sicheng Lu, Dong Tan et al., 2024)3Hu 2024 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)7Zhang 2025 A prescribed curvature flow on hyperbolic surfaces with infinite topological type (Xinrong Zhao and Puchun Zhou, 2025)6Zhao 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)4Ge 2025 The character of ideal circle patterns (Chang Li, Aijin Lin and Liangming Shen, 2026)5Li 2026
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 PDF

    Topics Discrete & graph Ricci, Other curvature flows · Shared terms “spherical background”, “background geometry”, “combinatorial curvature”

  2. 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 · Shared terms “spherical background”, “background geometry”, “combinatorial Ricci”

  3. 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 · Shared terms “background geometry”, “combinatorial Ricci”, “Chow Luo”

  4. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al. PDF

    Topics Discrete & graph Ricci · Same author Hao Yu · Shared terms “combinatorial Ricci”, “circle patterns”

  5. The character of ideal circle patterns

    Chang Li, Aijin Lin and Liangming Shen PDF

    Topics Discrete & graph Ricci · Same author Chang Li · Shared terms “circle patterns”, “cellular decomposition”, “combinatorial Ricci”

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

    Xinrong Zhao and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Other curvature flows · Shared terms “infinite cellular”, “cellular decomposition”, “background geometry”

  7. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang PDF

    Topics Discrete & graph Ricci, Other curvature flows · Shared terms “circle patterns”, “combinatorial”, “background geometry”

  8. Combinatorial Calabi flows for ideal circle patterns in spherical background geometry

    Ziping Lei and Puchun ZhouWider flows PDF

    Topics Other curvature flows · Shared terms “spherical background”, “background geometry”, “combinatorial”

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) 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) 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. 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.

  2. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.