Paper · May 31, 2018

Piecewise flat Ricci flow of compact without boundary three-manifolds

Rory Conboye

math.DGarXiv:1806.00095v2
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As on arXiv

Abstract

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homogeneity, and include Nil and Gowdy manifolds, a three-torus initially embedded in Euclidean four-space, and a perturbation of a flat three-torus. The piecewise flat Ricci flow of the first two are shown to converge to known smooth Ricci flow solutions, with the remaining two flowing asymptotically to flat metrics.

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Submitted
May 31, 2018
Last revised
Feb 23, 2024 (v2)
Subjects
math.DG
Ricci flow3-manifoldsNumerics & 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 Piecewise Flat Curvature and Ricci Flow in Three Dimensions (Rory Conboye and Warner A. Miller, 2016)2Conboye 2016 A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery (Paul M. Alsing, Warner A. Miller and Shing-Tung Yau, 2017)8Alsing 2017 How to see the eight Thurston geometries (Tiago Novello, Vinícius da Silva, Luiz Velho et al., 2020)7Novello 2020 Sphere Under Advection and Mean Curvature Flow (Bryce A. Besler, Tannis D. Kemp, Nils D. Forkert et al., 2020)5Besler 2020 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)1Conboye 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)3Garfinkle 2024 Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds (Miroslav Kolar and Daniel Sevcovic, 2025)4Kolar 2025 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)6Cochran 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization · Same author Rory Conboye · Shared terms “piecewise flat”, “flat Ricci”, “developed piecewise”

  2. Piecewise Flat Curvature and Ricci Flow in Three Dimensions

    Rory Conboye and Warner A. Miller

    Topics Ricci flow · Same author Rory Conboye · Shared terms “piecewise flat”, “flat Ricci”, “smooth Ricci”

  3. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Numerics & visualization · Shared terms “perturbation”, “numerical”

  4. Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

    Miroslav Kolar and Daniel SevcovicWider flows

    Topics Numerics & visualization, 3-manifolds · Shared terms “numerical”, “Euclidean”

  5. Sphere Under Advection and Mean Curvature Flow

    Bryce A. Besler, Tannis D. Kemp, Nils D. Forkert et al.Wider flows

    Topics Numerics & visualization · Shared terms “numerical”

  6. Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

    Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al.

    Topics Numerics & visualization

  7. How to see the eight Thurston geometries

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

    Topics Numerics & visualization, 3-manifolds

  8. A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

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

    Topics Numerics & visualization, 3-manifolds · Shared terms “piecewise”, “expected”, “numerical”

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) 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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

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.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.

  3. 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.