Paper · Oct 17, 2018

On -structures, special metrics and related flows

Marisa Fernández, Anna Fino, Alberto Raffero

math.DGarXiv:1810.07587
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Abstract

We review results about -structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.

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Submitted
Oct 17, 2018
Subjects
math.DG
DOI
10.1007/978-1-0716-0577-6_10

Comments: To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics"

Generalized & G₂ flowsRicci solitonsEinstein metricsHomogeneous spacesBoundaries & non-compact

Its neighbourhood

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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 Laplacian flow of homogeneous G2-structures and its solitons (Jorge Lauret, 2016)5Lauret 2016 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)2Panelli 2019 Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations (Peter Petersen and William Wylie, 2020)8Petersen 2020 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)1Podestà 2022 Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (Mark Haskins, Ilyas Khan and Alec Payne, 2022)7Haskins 2022 Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries (Fabio Podestà and Alberto Raffero, 2024)4Podestà 2024 Some remarks on strong G₂-structures with torsion (Anna Fino and Udhav Fowdar, 2025)6Fino 2025 Sasaki with torsion manifolds and string backgrounds (Beatrice Brienza, Anna Fino, Udhav Fowdar et al., 2026)3Brienza 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Alberto Raffero · Shared terms “compact homogeneous”

  2. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “compact homogeneous”

  3. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al.

    Topics Generalized & G₂ flows, Einstein metrics, Boundaries & non-compact · Same author Anna Fino

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

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Same author Alberto Raffero

  5. Laplacian flow of homogeneous G2-structures and its solitons

    Jorge Lauret

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “Laplacian flow”

  6. Some remarks on strong -structures with torsion

    Anna Fino and Udhav Fowdar

    Topics Generalized & G₂ flows · Same author Anna Fino · Shared terms “relation”

  7. Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

    Mark Haskins, Ilyas Khan and Alec Payne

    Topics Generalized & G₂ flows, Boundaries & non-compact, Ricci solitons · Shared terms “Laplacian flow”

  8. Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations

    Peter Petersen and William Wylie

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons

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.

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  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.