Paper · Aug 4, 2023

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

Xu Xu, Chao Zheng

math.DGarXiv:2308.02271
PDF Abstract page

As on arXiv

Abstract

Inversive distance circle packings introduced by Bowers-Stephenson are natural generalizations of Thurston's circle packings on surfaces. To find piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial -th Calabi flow for the Euclidean inversive distance circle packings. Due to the singularities possibly developed by these combinatorial curvature flows, the longtime existence and convergence of these combinatorial curvature flows have been a difficult problem for a long time. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Using the discrete conformal theory recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics on surfaces, we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. This provides effective algorithms for finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.

Other curvature flowsSingularitiesSurgeryGeometrizationSurfacesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stabilityDiscrete & 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 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)2Wu 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)3Xu 2022 Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary (Xu Xu and Chao Zheng, 2023)6Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)4Ba 2023 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)8Zhu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)5Xu 2023 A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces (Xu Xu and Chao Zheng, 2024)1Xu 2024 Discrete conformal structures on surfaces with boundary (III) – Deformation (Xu Xu and Chao Zheng, 2025)7Xu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

    Xu Xu and Chao Zheng PDF

    Topics Surgery, Surfaces, Kähler–Einstein & K-stability · Same author Chao Zheng · Shared terms “decorated piecewise”, “prescribed combinatorial”, “combinatorial curvature”

  2. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu PDF

    Topics Geometrization, Surgery, Discrete & graph Ricci · Shared terms “combinatorial Calabi”, “circle packings”, “fractional combinatorial”

  3. 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 “combinatorial curvature”, “circle packings”, “longtime existence”

  4. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Kähler–Einstein & K-stability, Long-time existence & convergence · Same author Chao Zheng · Shared terms “circle packings”, “combinatorial curvature”, “Calabi flow”

  5. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

    Topics Geometrization, Discrete & graph Ricci, Surfaces · Shared terms “combinatorial curvature”, “circle packings”, “distance circle”

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

  7. 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 “discrete conformal”, “combinatorial curvature”, “longtime existence”

  8. The existence of inversive distance circle packing on polyhedral surface

    Xiang Zhu PDF

    Topics Surgery, Discrete & graph Ricci, Singularities · Shared terms “distance circle”, “inversive distance”, “circle packings”

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.

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).