Paper · Apr 28, 2016

On the deformation of inversive distance circle packings

Huabin Ge, Wenshuai Jiang

math.GTarXiv:1604.08317
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Abstract

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flow can be used to deform any inversive distance circle packing to a unique packing with prescribed cone angle. We also give partial results on the range of all admissible cone angles, which generalize the classical Andreev-Thurston's theorem.

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Submitted
Apr 28, 2016
Subjects
math.GT, math-ph, math.DG

Comments: 25 pages

Ricci flowSingularitiesGeometrizationLong-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 discrete conformal factors on surfaces (Huabin Ge and Wenshuai Jiang, 2016)8Ge 2016 On the deformation of inversive distance circle packings, II (Huabin Ge and Wenshuai Jiang, 2016)1Ge 2016 On a combinatorial curvature for surfaces with inversive distance circle packing metrics (Huabin Ge and Xu Xu, 2017)2Ge 2017 On the deformation of ball packings (Huabin Ge, Wenshuai Jiang and Liangming Shen, 2018)4Ge 2018 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)3Xu 2023 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)5Zhu 2023 The existence of inversive distance circle packing on hyperbolic polyhedral surface (Xiang Zhu, 2023)7Zhu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)6Xu 2023
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, II

    Huabin Ge and Wenshuai Jiang

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

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

    Huabin Ge and Xu Xu

    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 Geometrization, Discrete & graph Ricci, Long-time existence & convergence · Shared terms “distance circle”, “inversive distance”, “circle packings”

  4. On the deformation of ball packings

    Huabin Ge, Wenshuai Jiang and Liangming ShenWider flows

    Topics Discrete & graph Ricci, Long-time existence & convergence, Singularities · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “develop singularities”, “combinatorial”, “exponentially fast”

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

    Xiang Zhu

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

  6. Deformation of discrete conformal structures on surfaces

    Xu Xu

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

  7. The existence of inversive distance circle packing on hyperbolic polyhedral surface

    Xiang Zhu

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

  8. On the deformation of discrete conformal factors on surfaces

    Huabin Ge and Wenshuai JiangWider flows

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “Luo”, “exponentially fast”, “converges exponentially”

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 can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.