Paper · Jul 24, 2025

Discrete conformal structures on surfaces with boundary (III) – Deformation

Xu Xu, Chao Zheng

math.DGarXiv:2507.18495
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Abstract

The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

Comments: arXiv admin note: text overlap with arXiv:2407.19501

Ricci flowOther curvature flowsSurfacesKähler geometryStability & uniquenessLong-time existence & convergenceDiscrete & graph Ricci

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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 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)6Wu 2021 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)7Luo 2021 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)2Xu 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)1Xu 2022 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)5Xu 2023 Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary (Xu Xu and Chao Zheng, 2023)3Xu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)4Xu 2023 A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces (Xu Xu and Chao Zheng, 2024)8Xu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Combinatorial curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Same author Chao Zheng · Shared terms “prescribed lengths”, “combinatorial curvature”, “geodesic boundaries”

  2. A new class of discrete conformal structures on surfaces with boundary

    Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Other curvature flows · Shared terms “conformal structures”, “discrete conformal”, “combinatorial”

  3. Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Same author Chao Zheng · Shared terms “combinatorial”, “longtime existence”, “Calabi flow”

  4. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “conformal structures”, “discrete conformal”, “combinatorial”

  5. Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Same author Chao Zheng · Shared terms “discrete conformal”, “combinatorial curvature”, “longtime existence”

  6. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “discrete conformal”, “conformal structures”, “combinatorial Calabi”

  7. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “combinatorial Calabi”, “geodesic boundaries”, “global convergence”

  8. A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

    Xu Xu and Chao Zheng PDF

    Topics Surfaces, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “hyperbolic metrics”, “combinatorial”, “longtime existence”

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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.

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) 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. 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.