Paper · Jul 5, 2025

Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

Aijin Lin, Longxiang Wu

math.DGWider flowsarXiv:2507.03901
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Abstract

Ge in his thesis introduced the combinatorial Calabi flows and established the long time existence and convergence of solutions to the flows in both hyperbolic and Euclidean background geometries. It is noteworthy that the existence of solutions to the combinatorial Calabi flows in hyperbolic background geometry proves to be more intricate and challenging compared to the Euclidean background geometry. The main difficulty is to establish the compactness, especially the lower boundeness along the flow equations. In this paper, we give two explicit estimates for the discrete Laplace operator based on the Glickenstein-Thomas formulation for discrete hyperbolic conformal structures. As applications, we give new proofs of the long time existence of solutions to the combinatorial Calabi flows established by Ge-Xu, Ge-Hua and the combinatorial -th Calabi flows established by Lin-Zhang in hyperbolic background geometry.

Other curvature flowsKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stability

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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 Calabi flows for ideal circle patterns in spherical background geometry (Ziping Lei and Puchun Zhou, 2023)1Lei 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)3Ba 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 Combinatorial Calabi flow for ideal circle pattern (Shengyu Li and Zhigang Wang, 2025)4Li 2025 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)5Zhang 2025 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)2Yang 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)7He 2025 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Kähler geometry · Shared terms “circle patterns”, “background geometry”, “combinatorial Calabi”

  2. Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

    Xiaorui Yang and Hao YuWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “circle patterns”, “Calabi flow”, “combinatorial”

  3. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “hyperbolic circle”, “Calabi flow”, “combinatorial Calabi”

  4. Combinatorial Calabi flow for ideal circle pattern

    Shengyu Li and Zhigang WangWider flows PDF

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

  5. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang PDF

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

  6. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Calabi flow”

  7. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

    Jie He and Haozhao LiWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Calabi flow”

  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 Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “background geometry”, “hyperbolic background”, “Calabi flow”

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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.

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.