Paper · May 27, 2018

3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

Huabin Ge, Bobo Hua

math.DGWider flowsarXiv:1805.10643
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Abstract

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the (extended) solid angle at each vertex is equal to 4π. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scaler curvature if the number of tetrahedra incident to each vertex is at most 22.

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Submitted
May 27, 2018
Subjects
math.DG, gr-qc, math.GT, math.MG

Comments: 34 pages

Yamabe flow3-manifoldsScalar curvatureLong-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 ball packings (Huabin Ge, Wenshuai Jiang and Liangming Shen, 2018)1Ge 2018 Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)4Feng 2020 Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)6Feng 2020 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)3Ge 2023 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)5Ge 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)7Ge 2025 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)8Ji 2025 Infinite Combinatorial Yamabe Flows in Three Dimensions (Bohao Ji, 2026)2Ji 2026
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 ball packings

    Huabin Ge, Wenshuai Jiang and Liangming ShenWider flows

    Topics Yamabe flow, Discrete & graph Ricci, 3-manifolds · Same author Huabin Ge · Shared terms “ball packing”, “combinatorial Yamabe”, “triangulation”

  2. Infinite Combinatorial Yamabe Flows in Three Dimensions

    Bohao JiWider flows

    Topics Yamabe flow, Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Yamabe”, “dimensional combinatorial”, “hyperbolic background”

  3. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “background geometry”, “combinatorial”, “hyperbolic background”

  4. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial”, “extended”, “exponentially fast”

  5. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci · Same authors Huabin Ge, Bobo Hua · Shared terms “hyperbolic background”, “background geometry”, “combinatorial”

  6. Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial”, “extended”, “flow converges”

  7. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al.

    Topics Discrete & graph Ricci, 3-manifolds · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial”

  8. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows

    Topics Yamabe flow, Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial Yamabe”, “background geometry”, “extended”

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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) 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. 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.

  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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.