Paper · Oct 27, 2023

On the Isometry Group of Immortal Homogeneous Ricci Flows

Roberto Araujo

math.DGarXiv:2310.18182v2The Journal of Geometric Analysis, volume 34, article number 173, (2024)
PDF Abstract page

As on arXiv

Abstract

We establish a structure result for the isometry group of non-compact, homogeneous manifolds admitting an immortal homogeneous Ricci flow solution.

Comments: 10 pages. Minor corrections made. Minor corrections and changes made to subsection 3.1. Final version as published

Ricci flowHomogeneous spacesBoundaries & non-compactLong-time existence & convergence

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 Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Carolyn Gordon and Michael Jablonski, 2021)6Gordon 2021 Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups (Victor Sanmartin-Lopez, 2022)8Sanmartin-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 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)1Araujo 2023 3-symmetric spaces, Ricci solitons, and homogeneous structures (Thomas Murphy and Paul-Andi Nagy, 2025)3Murphy 2025 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)2Araujo 2025 Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group (Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim, 2025)7Calvaruso 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)4Krishnan 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, Long-time existence & convergence, Boundaries & non-compact · Same author Roberto Araujo

  2. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo PDF

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

  3. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy PDF

    Topics Homogeneous spaces, Long-time existence & convergence, Boundaries & non-compact · Shared terms “isometry group”

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

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Homogeneous spaces, Long-time existence & convergence

  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. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski PDF

    Topics Homogeneous spaces · Shared terms “isometry group”

  7. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim PDF

    Topics Homogeneous spaces · Shared terms “isometry group”

  8. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez PDF

    Topics Homogeneous spaces, Boundaries & non-compact · Shared terms “isometry”

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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. 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.