Paper · Feb 4, 2020

Generalized Ricci flow on nilpotent Lie groups

Fabio Paradiso

math.DGarXiv:2002.01514v3
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Abstract

We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

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Submitted
Feb 4, 2020
Last revised
May 13, 2021 (v3)
Subjects
math.DG

Comments: 18 pages, to appear in Forum Mathematicum

Ricci flowGeneralized & G₂ flowsHomogeneous spaces

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  1. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “exact Courant”, “Courant algebroids”, “Lie groups”

  2. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “nilpotent Lie”, “Lie groups”, “left invariant”

  3. Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

    Mattia Pujia

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “nilpotent Lie”, “Lie groups”, “left invariant”

  4. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “nilpotent Lie”, “Heisenberg”, “connected nilpotent”

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

    Jorge Lauret

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “bracket flow”, “nilpotent Lie”, “Lie groups”

  6. Generalized scalar curvature and Ricci flow on compact Lie groups

    Zhiqi Chen, Jianhua Yu, Huibin Chen et al.

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Lie groups”, “nilpotent Lie”, “Garcia Fernandez”

  7. Courant Algebroid Relations, T-Dualities and Generalised Ricci Flow

    Thomas C. De Fraja, Vincenzo Emilio Marotta and Richard J. Szabo

    Topics Generalized & G₂ flows · Shared terms “Courant algebroids”, “exact Courant”, “explicit examples”

  8. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach

    Topics Generalized & G₂ flows · Shared terms “Courant algebroids”, “geometric flows”

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) 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

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    A computable laboratory: the flow on Thurston’s geometries.

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  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.