Paper · May 27, 2018

On the deformation of ball packings

Huabin Ge, Wenshuai Jiang, Liangming Shen

math.DGWider flowsarXiv:1805.10573
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we study the geometric aspects of ball packings on , where is a triangulation on a 3-manifold . We introduce a combinatorial Yamabe invariant , depending on the topology of and the combinatoric of . We prove that is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing Cooper-Rivin-Glickenstein functional. We then study the combinatorial Yamabe flow introduced by Glickenstein-. We first prove a small energy convergence theorem which says that the flow would converge to a constant curvature metric if the initial energy is close in a quantitative way to the energy of a constant curvature metric. We shall also prove: although the flow may develop singularities in finite time, there is a natural way to extend the solution of the flow so as it exists for all time. Moreover, if the triangulation is regular (that is, the number of tetrahedrons surrounding each vertex are all equal), then the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.

View PDF View HTML arXiv page
Submitted
May 27, 2018
Subjects
math.DG, math.GT
Yamabe flowSingularities3-manifoldsLong-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)2Ge 2016 On the deformation of inversive distance circle packings (Huabin Ge and Wenshuai Jiang, 2016)3Ge 2016 On the deformation of inversive distance circle packings, II (Huabin Ge and Wenshuai Jiang, 2016)4Ge 2016 On a combinatorial curvature for surfaces with inversive distance circle packing metrics (Huabin Ge and Xu Xu, 2017)8Ge 2017 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry (Huabin Ge and Bobo Hua, 2018)1Ge 2018 Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)6Feng 2020 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)7Ji 2025 Infinite Combinatorial Yamabe Flows in Three Dimensions (Bohao Ji, 2026)5Ji 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

    Huabin Ge and Bobo HuaWider flows

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

  2. On the deformation of discrete conformal factors on surfaces

    Huabin Ge and Wenshuai JiangWider flows

    Topics Yamabe flow, Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “constant curvature”, “triangulation”, “exponentially fast”

  3. On the deformation of inversive distance circle packings

    Huabin Ge and Wenshuai Jiang

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

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

    Huabin Ge and Wenshuai Jiang

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

  5. Infinite Combinatorial Yamabe Flows in Three Dimensions

    Bohao JiWider flows

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

  6. 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 author Huabin Ge · Shared terms “combinatorial”, “triangulation”, “exponentially fast”

  7. 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”, “triangulation”

  8. 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 “combinatorial”, “constant curvature”, “Yamabe”

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) 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) 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) 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.