Paper · Jan 7, 2017

On a combinatorial curvature for surfaces with inversive distance circle packing metrics

Huabin Ge, Xu Xu

math.GTarXiv:1701.01795v2
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Abstract

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to -curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture. We also use the combinatorial Ricci flow to study the corresponding -Yamabe problem.

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Submitted
Jan 7, 2017
Last revised
May 29, 2018 (v2)
Subjects
math.GT, math.DG
Journal
Journal of Functional Analysis 275 (2018) 523-558
Ricci flowScalar curvatureStability & uniquenessLong-time existence & convergenceDiscrete & 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 2016, 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 On the deformation of inversive distance circle packings (Huabin Ge and Wenshuai Jiang, 2016)1Ge 2016 On the deformation of inversive distance circle packings, II (Huabin Ge and Wenshuai Jiang, 2016)2Ge 2016 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)6Xu 2022 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)4Ge 2023 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)3Xu 2023 Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary (Xu Xu and Chao Zheng, 2023)8Xu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)7Xu 2023 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)5Ge 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the deformation of inversive distance circle packings

    Huabin Ge and Wenshuai Jiang

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same author Huabin Ge · Shared terms “distance circle”, “inversive distance”, “circle packings”

  2. On the deformation of inversive distance circle packings, II

    Huabin Ge and Wenshuai Jiang

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same author Huabin Ge · Shared terms “distance circle”, “inversive distance”, “circle packings”

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

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial curvature”, “distance circle”, “inversive distance”

  4. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same author Huabin Ge · Shared terms “combinatorial curvature”, “combinatorial Ricci”

  5. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Stability & uniqueness · Same author Huabin Ge · Shared terms “packing metrics”, “combinatorial Ricci”, “circle packings”

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

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial curvature”, “circle packings”, “triangulated surfaces”

  7. Deformation of discrete conformal structures on surfaces

    Xu Xu

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial curvature”, “triangulated surfaces”, “circle packings”

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

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Scalar curvature, Long-time existence & convergence · Shared terms “packing metrics”, “combinatorial”, “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.

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 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.

  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. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.