Paper · Nov 2, 2023

The existence of inversive distance circle packing on polyhedral surface

Xiang Zhu

math.DGarXiv:2311.01272
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Abstract

We prove that for any discrete curvature satisfying Gauss-Bonnet formula, there exist a unique up to scaling inversive distance circle packing in the discrete conformal equivalent class, whose polyhedral metric meets the target curvature. We prove it by constructing diffeomorphism between fiber bundles with cell decomposition based on Teichmüller spaces, and each discrete conformal equivalent class is a fiber passing through finite cell with respect to triangulations, which means we can do surgery on the discrete Ricci flow by edge flipping using a generalized Ptolemy equation to ensure it converge and never blow up.

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Submitted
Nov 2, 2023
Subjects
math.DG, math.GT, math.MG
Ricci flowSingularitiesSurgeryDiscrete & 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 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)8Perelman 2003 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)3Wu 2021 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)2Xu 2023 The existence of inversive distance circle packing on hyperbolic polyhedral surface (Xiang Zhu, 2023)1Zhu 2023 Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary (Guangming Hu, Yi Qi, Yu Sun et al., 2023)6Hu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)4Xu 2023 Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry (Guangming Hu, Sicheng Lu, Dong Tan et al., 2024)7Hu 2024 Evolution of weights on a connected finite graph (Jicheng Ma and Yunyan Yang, 2024)5Ma 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The existence of inversive distance circle packing on hyperbolic polyhedral surface

    Xiang Zhu

    Topics Surgery, Discrete & graph Ricci · Same author Xiang Zhu · Shared terms “distance circle”, “inversive distance”, “polyhedral surfaces”

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

    Xu Xu and Chao Zheng

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

  3. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu

    Topics Surgery, Discrete & graph Ricci, Singularities · Shared terms “discrete conformal”, “circle packings”, “polyhedral surfaces”

  4. Deformation of discrete conformal structures on surfaces

    Xu Xu

    Topics Discrete & graph Ricci · Shared terms “discrete conformal”, “circle packings”, “distance circle”

  5. Evolution of weights on a connected finite graph

    Jicheng Ma and Yunyan Yang

    Topics Surgery, Discrete & graph Ricci

  6. Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

    Guangming Hu, Yi Qi, Yu Sun et al.

    Topics Discrete & graph Ricci, Singularities · Shared terms “circle packings”, “decomposition”

  7. Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry

    Guangming Hu, Sicheng Lu, Dong Tan et al.

    Topics Discrete & graph Ricci · Shared terms “circle packings”

  8. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon

    Topics Surgery, Discrete & graph Ricci

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) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

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).