Paper · Mar 16, 2023

A combinatorial curvature flow in spherical background geometry

Huabin Ge, Bobo Hua, Puchun Zhou

math.GTarXiv:2303.09191
PDF Abstract page

As on arXiv

Abstract

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.

Comments: 11 pages, 2 figures

Ricci flowOther curvature flowsLong-time existence & convergenceDiscrete & graph Ricci

Its neighbourhood

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 for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)2Lei 2023 Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry (Guangming Hu, Ziping Lei, Yi Qi et al., 2024)8Hu 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)1Hu 2024 Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry (Guangming Hu, Sicheng Lu, Dong Tan et al., 2024)5Hu 2024 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)7Zhang 2025 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)6Ge 2025 Infinite combinatorial Ricci flow in spherical background geometry (Chang Li, Yangxiang Lu and Hao Yu, 2025)3Li 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
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Puchun Zhou · Shared terms “spherical background”, “background geometry”, “geodesic curvature”

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

    Ziping Lei and Puchun ZhouWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Same author Puchun Zhou · Shared terms “spherical background”, “background geometry”, “circle patterns”

  3. Infinite combinatorial Ricci flow in spherical background geometry

    Chang Li, Yangxiang Lu and Hao Yu PDF

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

  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 authors Huabin Ge, Bobo Hua, Puchun Zhou · Shared terms “circle patterns”, “combinatorial conditions”, “ideal circle”

  5. 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 · Same author Puchun Zhou · Shared terms “background geometry”, “hyperbolic background”, “combinatorial”

  6. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci · Same authors Huabin Ge, Bobo Hua, Puchun Zhou · Shared terms “background geometry”, “combinatorial”, “hyperbolic background”

  7. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang PDF

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

  8. 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, Other curvature flows · Same author Puchun Zhou · Shared terms “background geometry”, “geodesic curvature”, “total geodesic”

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) 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. 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.