Paper · Dec 17, 2021

Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

Carolyn Gordon, Michael Jablonski

math.DGarXiv:2112.09733v2
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Abstract

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to address these questions, partial results of Boehm, Lafuente, and Lauret are extended to show that left-invariant Ricci solitons on solvable Lie groups are unique up to scaling and isometry.

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Submitted
Dec 17, 2021
Last revised
Mar 13, 2023 (v2)
Subjects
math.DG

Comments: 31 pages; version 2 has improvements to the exposition

Ricci solitonsEinstein metricsHomogeneous spaces

Its neighbourhood

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  1. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “solvable Lie”, “left invariant”, “invariant Ricci”

  2. Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “solvmanifolds”, “invariant Ricci”, “left invariant”

  3. Kähler-Ricci solitons with almost maximal symmetry

    Ha Tuan Dung, Catherine Searle and Hung Tran

    Topics Homogeneous spaces, Ricci solitons · Shared terms “maximal symmetry”, “isometry group”

  4. Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

    Hamid Reza Salimi Moghaddam

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant Riemannian”

  5. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “expanding homogeneous”, “Lie groups”, “isometry”

  6. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson

    Topics Homogeneous spaces, Ricci solitons · Shared terms “solvmanifolds”, “Einstein”

  7. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “solvable Lie”, “left invariant”

  8. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces · Shared terms “isometry group”

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 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.