Paper · Sep 15, 2021

Four-dimensional generalized Ricci flows with nilpotent symmetry

Steven Gindi, Jeffrey Streets

math.DGarXiv:2109.07562
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Abstract

We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finite-dimensional family of solutions. The results are new for Ricci flow.

Ricci flowGeneralized & G₂ flows4-manifoldsHomogeneous spacesLong-time existence & convergence

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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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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

    Steven Gindi PDF

    Topics 4-manifolds, Homogeneous spaces · Same author Steven Gindi · Shared terms “nilpotent symmetry”, “blowdown limits”, “four dimensional”

  2. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  3. Topology of low-dimensional generalized Ricci solitons and string backgrounds

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “dimensional generalized”

  4. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Same author Jeffrey Streets · Shared terms “Lie”

  5. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang PDF

    Topics Generalized & G₂ flows, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “blowdown limits”

  6. Solvable Algebraic Generalized Ricci Solitons in low dimensions

    Lorenzo Mucciante PDF

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”, “Lie”

  7. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson PDF

    Topics 4-manifolds, Homogeneous spaces, Long-time existence & convergence · Shared terms “nilpotent symmetry”

  8. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

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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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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.