Paper · May 10, 2023

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

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

math.DGarXiv:2305.06119v3
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Abstract

In this note, we show that the classical Wallach manifold -admits metrics of positive intermediate Ricci curvature for that lose these properties under the homogeneous Ricci flow for . We make the same analyses to the family of Riemannian flag manifolds , concluding similar results. These explicitly verify some claims expected to be true among experts (see) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds.

Comments: Accepted version. To appear in the Special Issue of the Workshop on Geometric Structures and Moduli Spaces (WGSMS). Revista Matemática Contemporânea

Ricci flowHomogeneous spaces

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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.

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  1. The complete dynamics description of positively curved metrics in the Wallach flag manifold

    Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al. PDF

    Topics Ricci flow · Same authors Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins · Shared terms “curved metrics”, “positively curved”, “positive intermediate”

  2. Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

    Jason DeVito, David González-Álvaro and Masoumeh Zarei PDF

    Topics Homogeneous spaces · Shared terms “curved metrics”, “positively curved”

  3. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo PDF

    Topics Homogeneous spaces

  4. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces

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

    Mauro Patrão, Lucas Seco and Llohann D. Sperança PDF

    Topics Homogeneous spaces · Shared terms “invariant metrics”, “flag manifold”

  6. Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

    Fabio Podestà and Alberto Raffero PDF

    Topics Homogeneous spaces · Shared terms “positively curved”

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

    Roberto Araujo PDF

    Topics Homogeneous spaces

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

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Homogeneous spaces · Shared terms “invariant metrics”

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)

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.