Paper · Jun 2, 2016

Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

Renato G. Bettiol, Anusha M. Krishnan

math.DGarXiv:1606.00778v3
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As on arXiv

Abstract

We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.

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Submitted
Jun 2, 2016
Last revised
May 1, 2017 (v3)
Subjects
math.DG
Journal
Comm. Anal. Geom. 27 (2019), no. 3, 511-527

Comments: LaTeX2e, 11 pages, 2 figures, final version. To appear in Comm. Anal. Geom

Ricci flow4-manifoldsHomogeneous 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 2016, 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 flow does not preserve positive sectional curvature in dimension four

    Renato G. Bettiol and Anusha M. Krishnan

    Topics 4-manifolds, Homogeneous spaces · Same authors Renato G. Bettiol, Anusha M. Krishnan · Shared terms “sectional curvature”, “cohomogeneity”

  2. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”

  3. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao

    Shared terms “nonnegative sectional”, “sectional curvature”

  4. Classification of gradient steady Ricci solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics 4-manifolds · Shared terms “nonnegative sectional”, “sectional curvature”

  5. Four-dimensional Lorentzian algebraic Ricci solitons

    Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo

    Topics 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”

  6. -invariant steady gradient Ricci solitons on four-manifolds

    Timothy Buttsworth

    Topics 4-manifolds, Homogeneous spaces · Shared terms “cohomogeneity”, “four dimensional”

  7. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”

  8. Cohomogeneity two Ricci solitons with sub-Euclidean volume

    Benjy Firester and Raphael Tsiamis

    Topics 4-manifolds, Homogeneous spaces · Shared terms “cohomogeneity”, “four dimensional”

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… 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.