Paper · Nov 5, 2018

On the Ricci iteration for homogeneous metrics on spheres and projective spaces

Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein, Wolfgang Ziller

math.DGarXiv:1811.01724
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Abstract

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known results. The remaining and most challenging case is when the fibers are spheres of dimension 3. On the 3-sphere itself, using a result of Hamilton on the prescribed Ricci curvature equation, we establish existence and convergence of the Ricci iteration and confirm in this setting a conjecture on the relationship between ancient Ricci iterations and ancient solutions to the Ricci flow. In higher dimensions we obtain sufficient conditions for the solvability of the prescribed Ricci curvature equation as well as partial results on the behavior of the Ricci iteration.

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Submitted
Nov 5, 2018
Subjects
math.DG
Journal
Transformation Groups 26 (2021), 145--164
DOI
10.1007/s00031-020-09602-3

Comments: 17 pages

Ricci flowAncient solutions3-manifoldsHomogeneous spaces

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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 2016, 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 Ricci iteration on homogeneous spaces (Artem Pulemotov and Yanir A. Rubinstein, 2016)1Pulemotov 2016 Convergence of the Kähler-Ricci iteration (Tamás Darvas and Yanir A. Rubinstein, 2017)2Darvas 2017 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)6Brendle 2018 Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow (Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al., 2019)8Angenent 2019 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)3Sbiti 2020 SO(2)× SO(3)-invariant Ricci solitons and ancient flows on S⁴ (Timothy Buttsworth, 2021)5Buttsworth 2021 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)7Pediconi 2021 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)4Bourni 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci iteration on homogeneous spaces

    Artem Pulemotov and Yanir A. Rubinstein

    Topics Ancient solutions, Homogeneous spaces · Same authors Artem Pulemotov, Yanir A. Rubinstein · Shared terms “Ricci iteration”

  2. Convergence of the Kähler-Ricci iteration

    Tamás Darvas and Yanir A. Rubinstein

    Topics Ricci flow · Same author Yanir A. Rubinstein · Shared terms “Ricci iteration”

  3. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

    Topics Ancient solutions, Homogeneous spaces · Shared terms “homogeneous metrics”, “projective space”, “complex projective”

  4. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al.

    Topics Ancient solutions, 3-manifolds · Same author Timothy Buttsworth · Shared terms “Hamilton”

  5. -invariant Ricci solitons and ancient flows on

    Timothy Buttsworth

    Topics Ancient solutions · Same author Timothy Buttsworth

  6. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, 3-manifolds

  7. Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

    Francesco Pediconi and Sammy Sbiti

    Topics Ancient solutions, Homogeneous spaces · Shared terms “fibration”

  8. Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow

    Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al.

    Topics Ancient solutions, 3-manifolds

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.