Paper · Mar 10, 2016

Piecewise Flat Curvature and Ricci Flow in Three Dimensions

Rory Conboye, Warner A. Miller

math.DGarXiv:1603.03113
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Abstract

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat Ricci flow as a fractional rate of change of edge-lengths, again converging to the smooth Ricci flow for the manifolds tested.

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Submitted
Mar 10, 2016
Subjects
math.DG
Journal
Asian J. Math. Vol. 21, No. 6, pp. 1063-1098, December 2017
DOI
10.4310/AJM.2017.v21.n6.a30.4310/AJM.2017.v21.n6.a3
Ricci flowDiscrete & 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 A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery (Paul M. Alsing, Warner A. Miller and Shing-Tung Yau, 2017)3Alsing 2017 Piecewise flat Ricci flow of compact without boundary three-manifolds (Rory Conboye, 2018)2Conboye 2018 Geometric Sampling of Networks (Vladislav Barkanass, Jürgen Jost and Emil Saucan, 2020)6Barkanass 2020 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)1Conboye 2023 A Ricci flow on graphs from effective resistance (Aleyah Dawkins, Vishal Gupta, Mark Kempton et al., 2024)8Dawkins 2024 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)5Bai 2025 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)7Hua 2026 Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow (Soichiro Uemura, 2026)4Uemura 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 Ricci flow · Same author Rory Conboye · Shared terms “piecewise flat”, “flat Ricci”, “smooth Ricci”

  2. Piecewise flat Ricci flow of compact without boundary three-manifolds

    Rory Conboye

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

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

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

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

  4. Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

    Soichiro Uemura

    Topics Discrete & graph Ricci · Shared terms “triangulation”

  5. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al.

    Topics Discrete & graph Ricci

  6. Geometric Sampling of Networks

    Vladislav Barkanass, Jürgen Jost and Emil Saucan

    Topics Discrete & graph Ricci · Shared terms “sampling”

  7. The Ollivier Ricci flow with prescribed curvature on infinite graphs

    Bobo Hua, Yong Lin and Shuang Liu

    Topics Discrete & graph Ricci

  8. A Ricci flow on graphs from effective resistance

    Aleyah Dawkins, Vishal Gupta, Mark Kempton et al.

    Topics Discrete & graph 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) 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. 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. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.