Paper · Jun 22, 2023

Stability of piecewise flat Ricci flow

Rory Conboye

math.DGarXiv:2306.12745
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Abstract

The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability. These adaptations have also been used in a related paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.

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Submitted
Jun 22, 2023
Subjects
math.DG
Ricci flowStability & uniquenessNumerics & 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 2021, 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 numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)3Garfinkle 2022 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)1Garfinkle 2024 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)2Bahadursingh 2024 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)4He 2025 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)6Bai 2025 Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow (Jihui Wang, Matteo Lulli and Antonino Marciano, 2026)5Wang 2026 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)7Cochran 2026 Using Ricci Flow in the Construction of Homogeneous Cosmological Models (Fan Zhang and Lee Lindblom, 2026)8Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”

  2. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”

  3. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “stability analysis”, “numerical simulations”, “adaptation”

  4. A curvature flow approach to dorsal closure modelling

    Shuhui He, Ben Whale, Glen Wheeler et al.Wider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical”, “simulations”, “approximation”

  5. Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

    Jihui Wang, Matteo Lulli and Antonino Marciano

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “simulations”, “numerical”

  6. Ricci Flow on Weighted Digraphs with Balancing Factor

    Shuliang Bai, Rui Li, Shuang Liu et al.

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical”

  7. 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

  8. Using Ricci Flow in the Construction of Homogeneous Cosmological Models

    Fan Zhang and Lee Lindblom

    Topics Numerics & visualization · Shared terms “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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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. 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.