Paper · Oct 16, 2021

A new class of discrete conformal structures on surfaces with boundary

Xu Xu

math.GTarXiv:2110.08474v2Calc. Var. Partial Differential Equations 61 (2022), no. 4, Paper No. 141
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Abstract

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.

Comments: 28 pages, 5 figures, title changed, references are updated

Ricci flowYamabe flowOther curvature flows3-manifoldsSurfacesKähler geometryStability & uniquenessDiscrete & 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)3Wu 2021 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)4Luo 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)5Xu 2022 Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary (Xu Xu and Chao Zheng, 2023)7Xu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)2Xu 2023 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)6Xiong 2025 Infinite combinatorial Ricci flow in spherical background geometry (Chang Li, Yangxiang Lu and Hao Yu, 2025)8Li 2025 Discrete conformal structures on surfaces with boundary (III) – Deformation (Xu Xu and Chao Zheng, 2025)1Xu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Discrete conformal structures on surfaces with boundary (III) – Deformation

    Xu Xu and Chao Zheng PDF

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

  2. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

    Topics Yamabe flow, Discrete & graph Ricci, Surfaces · Shared terms “conformal structures”, “discrete conformal”, “Luo combinatorial”

  3. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu PDF

    Topics Yamabe flow, Discrete & graph Ricci, Surfaces · Shared terms “discrete conformal”, “conformal structures”, “Luo combinatorial”

  4. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Yamabe flow, Discrete & graph Ricci, Surfaces · Shared terms “combinatorial”, “geodesic boundaries”, “Guo”

  5. Combinatorial curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Other curvature flows · Shared terms “combinatorial curvature”, “prescribed lengths”, “geodesic boundaries”

  6. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Other curvature flows · Shared terms “Luo”, “Guo”, “combinatorial”

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

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, 3-manifolds · Shared terms “combinatorial”, “Guo Luo”, “Calabi flow”

  8. 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 “combinatorial Ricci”, “combinatorial curvature”, “Chow Luo”

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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) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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. 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.

  3. 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.