Paper · Jul 7, 2025

Finite extinction time of a family of homogeneous Ricci flows

Roberto Araujo

math.DGarXiv:2507.05097v2
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Abstract

We show that for a broad family of noncompact homogeneous Riemannian manifolds, the corresponding homogeneous Ricci flow solutions have finite extinction time, thereby confirming the dynamical Alekseevskii conjecture for these spaces. As an application, we prove that on such homogeneous manifolds , the space of all -invariant positive scalar curvature metrics is contractible.

Comments: 28 pages. Minor revisions. Accepted by Mathematische Zeitschrift

Ricci flowScalar curvatureHomogeneous spacesBoundaries & non-compact

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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 Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (Grisha Perelman, 2003)4Perelman 2003 On the embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)8Patrão 2021 Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups (Victor Sanmartin-Lopez, 2022)6Sanmartin-Lo 2022 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)5Cavenaghi 2023 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)2Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)1Araujo 2023 The dynamical Alekseevskii conjecture in dimension five (Maher Billon, 2025)3Billon 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)7Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces, Boundaries & non-compact · Same author Roberto Araujo · Shared terms “finite extinction”, “extinction time”, “Alekseevskii Conjecture”

  2. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo PDF

    Topics Homogeneous spaces, Boundaries & non-compact · Same author Roberto Araujo

  3. The dynamical Alekseevskii conjecture in dimension five

    Maher Billon PDF

    Topics Homogeneous spaces · Shared terms “Alekseevskii Conjecture”, “Dynamical Alekseevskii”

  4. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Ricci flow · Shared terms “finite extinction”, “extinction time”

  5. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins PDF

    Topics Homogeneous spaces

  6. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez PDF

    Topics Homogeneous spaces, Scalar curvature, Boundaries & non-compact

  7. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Homogeneous spaces

  8. On the embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança PDF

    Topics Homogeneous spaces

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.