Paper · Jul 4, 2016

On the deformation of inversive distance circle packings, II

Huabin Ge, Wenshuai Jiang

math.GTarXiv:1607.00833
View PDF View HTML arXiv page

As on arXiv

Abstract

We show that the results in are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston's theorem to some extent.

View PDF View HTML arXiv page
Submitted
Jul 4, 2016
Subjects
math.GT, math.DG
Journal
J. Funct. Anal. 272. no. 9, 3573-3595, 2017
DOI
10.1016/j.jfa.2016.12.021

Comments: Comments are welcome

Ricci flowGeometrizationLong-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 discrete conformal factors on surfaces (Huabin Ge and Wenshuai Jiang, 2016)5Ge 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)6Ge 2018 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)7Ge 2023 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)8Zhu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)4Xu 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

    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. 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”

  5. 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”, “extended”

  6. On the deformation of ball packings

    Huabin Ge, Wenshuai Jiang and Liangming ShenWider flows

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

  7. 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 “background geometry”, “hyperbolic background”, “combinatorial”

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

    Xiang Zhu

    Topics Discrete & graph Ricci · 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.

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) 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) 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. 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.