Paper · Jun 15, 2018

Convergence of combinatorial Ricci flows to degenerate circle patterns

Asuka Takatsu

math.DGarXiv:1806.05812v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.

View PDF View HTML arXiv page
Submitted
Jun 15, 2018
Last revised
Aug 19, 2018 (v2)
Subjects
math.DG, math.GT
Ricci flowSurfacesDiscrete & 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 Circle patterns on surfaces of finite topological type (Huabin Ge, Bobo Hua and Ze Zhou, 2019)1Ge 2019 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)5Ge 2023 The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry (Guangming Hu, Ziping Lei, Yu Sun et al., 2024)6Hu 2024 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)4Zhang 2025 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)8Ge 2025 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)2Xiong 2025 Infinite combinatorial Ricci flow in spherical background geometry (Chang Li, Yangxiang Lu and Hao Yu, 2025)7Li 2025 Branched α-combinatorial Ricci flows on closed surfaces with Euler characteristic χ≤ 0 (Wenjun Li, Rongyuan Liu, Guohao Chen et al., 2025)3Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Circle patterns on surfaces of finite topological type

    Huabin Ge, Bobo Hua and Ze Zhou

    Topics Discrete & graph Ricci, Surfaces · Shared terms “circle patterns”, “combinatorial Ricci”, “Chow”

  2. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu

    Topics Discrete & graph Ricci, Surfaces · Shared terms “circle patterns”, “combinatorial Ricci”, “Luo”

  3. Branched -combinatorial Ricci flows on closed surfaces with Euler characteristic

    Wenjun Li, Rongyuan Liu, Guohao Chen et al.

    Topics Discrete & graph Ricci, Surfaces · Shared terms “combinatorial Ricci”, “Euler characteristic”

  4. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang

    Topics Discrete & graph Ricci · Shared terms “circle patterns”, “combinatorial”

  5. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci · Shared terms “circle patterns”, “combinatorial”, “Chow”

  6. The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry

    Guangming Hu, Ziping Lei, Yu Sun et al.

    Topics Discrete & graph Ricci · Shared terms “circle patterns”, “combinatorial Ricci”

  7. Infinite combinatorial Ricci flow in spherical background geometry

    Chang Li, Yangxiang Lu and Hao Yu

    Topics Discrete & graph Ricci · Shared terms “combinatorial Ricci”, “circle patterns”, “Chow”

  8. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Surfaces · Shared terms “combinatorial 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 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.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.