Generalized Ricci flow on aligned homogeneous spaces

Valeria Gutiérrez

Paper · Jan 6, 2024

Generalized Ricci flow on aligned homogeneous spaces

Valeria Gutiérrez

math.DGarXiv:2401.03332

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Abstract

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g, H)$ on a manifold $M$ , where $g$ is a Riemannian metric and $H$ a closed $3$ -form, such that $H$ is $g$ -harmonic and $Rc (g) = \frac{1}{4} H_{g}^{2}$ . Given two standard Einstein homogeneous spaces $G_{i} / K$ , where each $G_{i}$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M = G_{1} \times G_{2} /Δ K$ . Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$ -invariant metrics and, if $G_{1} = G_{2}$ , then it is globally stable.

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Submitted: Jan 6, 2024
Subjects: math.DG
Journal: Proceedings of the Edinburgh Mathematical Society 68 (2025) 226-246
DOI: 10.1017/S0013091524000658

Ricci flow Generalized & G₂ flows Ricci-flat & Calabi–Yau Homogeneous spaces

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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) 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

11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
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.
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.

[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.

Generalized Ricci flow on aligned homogeneous spaces

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