Paper · May 26, 2025

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

Xinrong Zhao, Puchun Zhou

math.GTarXiv:2505.20091
PDF Abstract page

As on arXiv

Abstract

In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

Comments: 22 pages, 4 figures

Ricci flowOther curvature flowsSurfacesBoundaries & non-compactDiscrete & 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 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)6Ge 2023 Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary (Guangming Hu, Yi Qi, Yu Sun et al., 2023)1Hu 2023 Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry (Guangming Hu, Ziping Lei, Yi Qi et al., 2024)2Hu 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)4Hu 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 Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)3Ge 2025 Infinite combinatorial Ricci flow in spherical background geometry (Chang Li, Yangxiang Lu and Hao Yu, 2025)7Li 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)8Ge 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 · Same author Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “cellular decomposition”

  2. 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, Surfaces, Other curvature flows · Same author Puchun Zhou · Shared terms “total geodesic”, “geodesic curvature”, “prescribed total”

  3. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Surfaces · Same author Puchun Zhou · Shared terms “hyperbolic background”, “background geometry”, “packing metrics”

  4. 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 “total geodesic”, “prescribed total”, “geodesic curvature”

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

  6. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

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

  7. 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 “infinite cellular”, “cellular decomposition”, “background geometry”

  8. 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, Boundaries & non-compact · Same author Puchun Zhou · Shared terms “discrete analogue”, “proposed”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.