Paper · Oct 19, 2026

Using Ricci Flow in the Construction of Homogeneous Cosmological Models

Fan Zhang, Lee Lindblom

gr-qcarXiv:2610.27340
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Abstract

This study develops numerical methods for computing homogeneous and isotropic cosmological models on manifolds with non-trivial spatial topologies. The use of Ricci flow is introduced here to aid in the construction of globally homogeneous initial data on these manifolds. The initial data produced in this way are evolved by solving Einstein's equation numerically to construct homogeneous and isotropic cosmological models on three exemplar manifolds in this study: Seifert-Weber space, the lens space , and Sixth-Turn space.

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Submitted
Oct 19, 2026
Subjects
gr-qc

Comments: 13 Pages, 20 Figures

Ricci flowHomogeneous spacesPhysicsNumerics & visualization

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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 Kähler-Ricci flow on rational homogeneous varieties (Eder Correa, 2021)3Correa 2021 Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes (Davide De Biasio, Julian Freigang, Dieter Lust et al., 2022)4De Biasio 2022 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)5Conboye 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)8Garfinkle 2024 Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow (Jihui Wang, Matteo Lulli and Antonino Marciano, 2026)2Wang 2026 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)1Cochran 2026 Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry (Yeong-Bok Bae, Young-Hwan Hyun and Gungwon Kang, 2026)7Bae 2026 Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory (G. Bonelli, S. Lacroix, T. Pedroni et al., 2026)6Bonelli 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Homogeneous spaces

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

    Jihui Wang, Matteo Lulli and Antonino Marciano

    Topics Numerics & visualization, Physics · Shared terms “cosmological”, “numerical”

  3. Kähler-Ricci flow on rational homogeneous varieties

    Eder Correa

    Topics Numerics & visualization, Homogeneous spaces · Shared terms “numerical”

  4. Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

    Davide De Biasio, Julian Freigang, Dieter Lust et al.

    Topics Numerics & visualization, Physics · Shared terms “numerically”, “Einstein”

  5. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization · Shared terms “numerical”

  6. Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

    G. Bonelli, S. Lacroix, T. Pedroni et al.

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

  7. Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry

    Yeong-Bok Bae, Young-Hwan Hyun and Gungwon Kang

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

  8. 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 “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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  3. 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.