Paper · Mar 9, 2024

Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry

Guangming Hu, Ziping Lei, Yi Qi, Puchun Zhou

math.DGWider flowsarXiv:2403.05777v2
PDF Abstract page

As on arXiv

Abstract

In hyperbolic background geometry, we investigate a generalized circle packing (including circles, horocycles and hypercycles) with conical singularities on a surface with boundary, which has a total geodesic curvature on each generalized circle of this circle packing and a discrete Gaussian curvature on the center of each dual circle. The purpose of this paper is to find this type of circle packings with prescribed total geodesic curvatures on generalized circles and discrete Gaussian curvatures on centers of dual circles. To achieve this goal, we firstly establish existence and rigidity on this type of circle packings by the variational principle. Secondly, for , we introduce combinatorial -th Calabi flows to find the circle packing with prescribed total geodesic curvatures on generalized circles and discrete Gaussian curvatures on centers of dual circles for the first time.

Comments: 30 pages, 10 figures

Other curvature flowsSingularitiesSurfacesKähler geometryStability & uniquenessKähler–Einstein & K-stabilityDiscrete & 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)4Lei 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)6Ba 2023 Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary (Guangming Hu, Yi Qi, Yu Sun et al., 2023)1Hu 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)5Hu 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 Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings (Guangming Hu, Ziping Lei, Yanlin Li et al., 2024)2Hu 2024 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)8Ge 2025 A prescribed curvature flow on hyperbolic surfaces with infinite topological type (Xinrong Zhao and Puchun Zhou, 2025)7Zhao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

    Guangming Hu, Yi Qi, Yu Sun et al. PDF

    Topics Discrete & graph Ricci, Surfaces, Singularities · Same authors Guangming Hu, Yi Qi, Puchun Zhou · Shared terms “total geodesic”, “circle packings”, “geodesic curvature”

  2. 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, Other curvature flows, Kähler geometry · Same authors Guangming Hu, Ziping Lei · Shared terms “circle packings”, “generalized circle”, “hyperbolic background”

  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, Stability & uniqueness · Same authors Guangming Hu, Puchun Zhou · Shared terms “circle packings”, “hyperbolic background”, “background geometry”

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

    Ziping Lei and Puchun ZhouWider flows PDF

    Topics Other curvature flows, Kähler geometry · Same authors Ziping Lei, Puchun Zhou · Shared terms “background geometry”, “total geodesic”, “Calabi flow”

  5. 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, Ziping Lei, Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “background geometry”

  6. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Other curvature flows · Shared terms “circle packings”, “generalized circle”, “total geodesic”

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

    Xinrong Zhao and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Surfaces, Other curvature flows · Same author Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “prescribed total”

  8. 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 “hyperbolic background”, “background geometry”, “circle packings”

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.

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) 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) 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. 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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.