Paper · Nov 8, 2023

The existence of inversive distance circle packing on hyperbolic polyhedral surface

Xiang Zhu

math.DGarXiv:2311.04450
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Abstract

In this paper, we prove that given a hyperbolic polyhedral metric with an inversive distance circle packing, and a target discrete curvature satisfying Gauss-Bonnet formula, there exist a unique inversive distance circle packing which is discrete conformal to the former one. We deform the surface by discrete Ricci flow, and do surgery by edge flipping when the orthogonal circles of some faces are about to be non-compact. The revised weighted Delaunay inequality of hyperbolic case implies the compactness of the orthogonal circle. We use a variational principle of a convex Ricci potential defined on the fiber bundles with cell-decomposition and differential structure based on Teichmüller space to finish the proof.

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Submitted
Nov 8, 2023
Subjects
math.DG, math.GT, math.MG

Comments: arXiv admin note: substantial text overlap with arXiv:2311.01272

Ricci flowSurgeryLimit spaces & RCDBoundaries & non-compactDiscrete & 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 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)5Hu 2024 Evolution of weights on a connected finite graph (Jicheng Ma and Yunyan Yang, 2024)7Ma 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 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 · 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 · Shared terms “circle packings”, “polyhedral surfaces”, “discrete conformal”

  4. Deformation of discrete conformal structures on surfaces

    Xu Xu

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

  5. 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”

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

    Guangming Hu, Yi Qi, Yu Sun et al.

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

  7. Evolution of weights on a connected finite graph

    Jicheng Ma and Yunyan Yang

    Topics Surgery, Discrete & graph Ricci

  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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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

  3. 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.