Paper · Feb 17, 2025

Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons

M. M. Akbar, M. Self

gr-qcarXiv:2502.11428
View PDF View HTML arXiv page

As on arXiv

Abstract

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in dimensions is invariant under the action of a one-parameter Lie group, and the group action on any metric can be expressed in a closed, universal form. This enables the generation of a one-parameter family of solutions from any given "seed" solution of the system without solving additional equations, as well as one-parameter families of local steady Ricci solitons. This extends the Lie point symmetry in four dimensions, found earlier for axisymmetric static vacuum systems, and provides the first example of solution generation in higher-dimensional vacuum Einstein equations that can be realized purely algebraically.

View PDF View HTML arXiv page
Submitted
Feb 17, 2025
Subjects
gr-qc

Comments: 11 pages, no figures

Ricci solitonsRicci-flat & Calabi–YauHomogeneous spaces

Its neighbourhood

Connected papers

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 2021, 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 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)5Podestà 2022 Symmetries of Ricci Flows (Enrique López, Stylianos Dimas and Yuri Bozhkov, 2022)3López 2022 Inhomogeneous deformations of Einstein solvmanifolds (Adam Thompson, 2023)6Thompson 2023 Generalized Ricci flow on aligned homogeneous spaces (Valeria Gutiérrez, 2024)2Gutiérrez 2024 Homogeneous Generalized Ricci flows II (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2026)1Fusi 2026 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)7Foscolo 2026 HKT geometry on locally conformally hyperkähler manifolds (Giovanni Gentili and Luigi Vezzoni, 2026)8Gentili 2026 Two-step nilpotent Lie algebras obtained by quivers and geometric structures (Fumika Mizoguchi, 2026)4Mizoguchi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces, Ricci solitons · Shared terms “flat metrics”, “Lie groups”, “Killing”

  2. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces · Shared terms “flat metrics”, “Lie groups”, “Einstein”

  3. Symmetries of Ricci Flows

    Enrique López, Stylianos Dimas and Yuri Bozhkov

    Shared terms “Lie point”, “point symmetries”, “Einstein equations”

  4. Two-step nilpotent Lie algebras obtained by quivers and geometric structures

    Fumika Mizoguchi

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces, Ricci solitons · Shared terms “Lie”

  5. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces

  6. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson

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

  7. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces, Ricci solitons · Shared terms “parameter family”, “Einstein”

  8. HKT geometry on locally conformally hyperkähler manifolds

    Giovanni Gentili and Luigi Vezzoni

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces

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) 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

  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.