Paper · Sep 3, 2023

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

Xu Xu, Chao Zheng

math.DGarXiv:2309.01205
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Abstract

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

Comments: To appear in CVPDE

Ricci flowOther curvature flows3-manifoldsSurfacesKähler geometryScalar curvatureStability & uniquenessBoundaries & non-compactLong-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 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)6Xu 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)4Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)3Ba 2023 Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary (Guangming Hu, Yi Qi, Yu Sun et al., 2023)8Hu 2023 A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces (Xu Xu and Chao Zheng, 2024)7Xu 2024 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)5Xiong 2025 Discrete conformal structures on surfaces with boundary (III) – Deformation (Xu Xu and Chao Zheng, 2025)2Xu 2025
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 “generalized circle”, “circle packings”, “combinatorial”

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

    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”

  3. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “circle packings”, “combinatorial”, “generalized circle”

  4. 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 “combinatorial curvature”, “circle packings”, “prescribed combinatorial”

  5. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “generalized circle”, “Guo Luo”, “Luo generalized”

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

    Xu Xu PDF

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

  7. 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 “combinatorial”, “longtime existence”, “Calabi flow”

  8. 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, Stability & uniqueness · Shared terms “circle packings”, “generalized circle”, “Motivated”

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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) 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) 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)

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.