Paper · Dec 12, 2020

On the dynamical behaviour of the generalized Ricci flow

Alberto Raffero, Luigi Vezzoni

math.DGarXiv:2012.06792
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Abstract

Motivated by Müller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.

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Submitted
Dec 12, 2020
Subjects
math.DG
Journal
The journal of Geometric Analysis, Vol. 31, pp. 10498-10509 (2021)
DOI
10.1007/s12220-021-00656-7

Comments: 10 pages

Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauStability & uniqueness

Its neighbourhood

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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.

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 Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow (Alix Deruelle and Tristan Ozuch, 2021)6Deruelle 2021 The Stability of Generalized Ricci Solitons (Kuan-Hui Lee, 2022)5Lee 2022 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)1Podestà 2022 The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds (Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni, 2023)8Bedulli 2023 Generalized Ricci flow on aligned homogeneous spaces (Valeria Gutiérrez, 2024)4Gutiérrez 2024 Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries (Fabio Podestà and Alberto Raffero, 2024)7Podestà 2024 Dynamical stability of Pluriclosed and Generalized Ricci solitons (Kuan-Hui Lee, 2025)2Lee 2025 Homogeneous Generalized Ricci flows II (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2026)3Fusi 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, Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Alberto Raffero

  2. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Stability & uniqueness · Shared terms “dynamical stability”

  3. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “flat metrics”, “dynamical stability”

  4. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau · Shared terms “flat metrics”

  5. The Stability of Generalized Ricci Solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Stability & uniqueness · Shared terms “dynamical stability”, “Haslhofer”

  6. Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

    Alix Deruelle and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “flat metrics”, “dynamical”, “instability”

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

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows · Same author Alberto Raffero

  8. The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

    Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau · Same author Luigi Vezzoni

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) 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-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.