Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

Rohollah Bakhshandeh-Chamazkoti

Paper · Aug 25, 2021

Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

Rohollah Bakhshandeh-Chamazkoti

math.DGarXiv:2108.11380

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Abstract

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_{3} \times R$ and $G_{4}$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_{3} \times R$ , $g_{λ}^{+}$ is a shrinking while $g_{λ}^{+}$ and $g_{μ}$ are expanding and also $g_{0}^{1}, g_{0}^{2}, g_{0}^{3}$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_{4}$ only $g_{1}^{λ}, g_{2}^{λ}$ have Lorentz Ricci solitons and $g_{2}^{λ}$ is a shrinking.

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Submitted: Aug 25, 2021
Subjects: math.DG

Ricci solitons 4-manifolds Homogeneous spaces Physics

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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 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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
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.

[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.

Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

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