Paper · Sep 21, 2017

A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

Paul M. Alsing, Warner A. Miller, Shing-Tung Yau

math.DGarXiv:1709.08494
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Abstract

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure – a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

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Submitted
Sep 21, 2017
Subjects
math.DG, gr-qc

Comments: 8 pages, 6 figures. arXiv admin note: text overlap with arXiv:1308.4148

Ricci flowSingularitiesSurgery(Non)collapsingGeometrization3-manifoldsRotational symmetry & warped productsDiscrete & graph RicciNumerics & visualization

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 The entropy formula for the Ricci flow and its geometric applications (Grisha Perelman, 2002)4Perelman 2002 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)7Morgan 2008 Piecewise Flat Curvature and Ricci Flow in Three Dimensions (Rory Conboye and Warner A. Miller, 2016)1Conboye 2016 How to see the eight Thurston geometries (Tiago Novello, Vinícius da Silva, Luiz Velho et al., 2020)2Novello 2020 Ollivier Ricci-flow on weighted graphs (Shuliang Bai, Yong Lin, Linyuan Lu et al., 2020)5Bai 2020 Geometric Sampling of Networks (Vladislav Barkanass, Jürgen Jost and Emil Saucan, 2020)6Barkanass 2020 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)8Zhu 2023 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)3Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Piecewise Flat Curvature and Ricci Flow in Three Dimensions

    Rory Conboye and Warner A. Miller

    Topics Discrete & graph Ricci · Same author Warner A. Miller · Shared terms “edge lengths”, “simplicial”, “piecewise”

  2. How to see the eight Thurston geometries

    Tiago Novello, Vinícius da Silva, Luiz Velho et al.

    Topics Geometrization, Numerics & visualization, 3-manifolds · Shared terms “Thurston”

  3. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “Thurston geometrization”, “Hamilton”

  4. The entropy formula for the Ricci flow and its geometric applications

    Grisha PerelmanThe canon

    Topics Geometrization, (Non)collapsing, 3-manifolds · Shared terms “Thurston geometrization”, “expression”, “Hamilton”

  5. Ollivier Ricci-flow on weighted graphs

    Shuliang Bai, Yong Lin, Linyuan Lu et al.

    Topics Discrete & graph Ricci · Same author Shing-Tung Yau

  6. Geometric Sampling of Networks

    Vladislav Barkanass, Jürgen Jost and Emil Saucan

    Topics Numerics & visualization, Discrete & graph Ricci · Shared terms “Forman”

  7. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon

    Topics Geometrization, Surgery, Discrete & graph Ricci

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

    Xiang Zhu

    Topics Surgery, Discrete & graph Ricci, Singularities · Shared terms “decomposition”, “constructing”

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 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.

  2. 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.

  3. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

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