Paper · Apr 15, 2025

Prescribing hyperbolic bordered surfaces via combinatorial flows

Shengyu Li, Zhi-Gang Wang

math.CVWider flowsarXiv:2504.11266v2
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Abstract

The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that generates a hyperbolic surface whose lengths of boundary components are prescribed positive numbers. Furthermore, a generalized combinatorial Yamabe flow is introduced in the same geometry setting, with the long time existence and convergence established. This result yields an algorithm for searching bordered surfaces, which may accelerate convergence speed.

Comments: Some of the results overlap with previous findings

Yamabe flowOther curvature flowsKähler geometryLong-time existence & convergence

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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 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)8Wu 2021 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)2Luo 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)1Li 2022 Convergence rate of the Q-curvature flow (Pak Tung Ho and Sanghoon Lee, 2024)4Ho 2024 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)6Lye 2024 Combinatorial Calabi flow for ideal circle pattern (Shengyu Li and Zhigang Wang, 2025)3Li 2025 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)7Ji 2025 Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces (Yuerong Bian and Xu Xu, 2026)5Bian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence · Same author Shengyu Li · Shared terms “bordered surfaces”, “hyperbolic bordered”, “combinatorial Yamabe”

  2. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence, Other curvature flows · Shared terms “combinatorial”, “Calabi flow”, “hyperbolic surface”

  3. Combinatorial Calabi flow for ideal circle pattern

    Shengyu Li and Zhigang WangWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows, Kähler geometry · Same author Shengyu Li · Shared terms “combinatorial”, “Calabi flow”, “converges exponentially”

  4. Convergence rate of the -curvature flow

    Pak Tung Ho and Sanghoon LeeWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence, Other curvature flows · Shared terms “converges exponentially”

  5. Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

    Yuerong Bian and Xu XuWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence · Shared terms “combinatorial Yamabe”, “hyperbolic surface”

  6. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence

  7. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence · Shared terms “combinatorial Yamabe”

  8. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu PDF

    Topics Yamabe flow, Long-time existence & convergence, Other curvature flows · Shared terms “fractional combinatorial”, “combinatorial Calabi”, “Calabi flow”

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) 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.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.