Paper · Sep 10, 2026

Harmonicity and Existence of Algebraic Generalized Ricci Solitons

Huibin Chen, Zhiqi Chen, Fuhai Zhu

math.DGarXiv:2609.16029
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Abstract

In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions for . Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra , , admits a generalized nilsoliton if and only if .

Generalized & G₂ flowsRicci solitonsHomogeneous spacesEigenvalues & spectrum

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  1. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu PDF

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Same authors Huibin Chen, Zhiqi Chen, Fuhai Zhu · Shared terms “algebraic generalized”, “harmonic torsion”, “generalized nilsolitons”

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    Topics Generalized & G₂ flows, Homogeneous spaces · Same authors Huibin Chen, Zhiqi Chen · Shared terms “Lie algebras”, “nilpotent Lie”, “Killing”

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

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    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “nilpotent Lie”, “step nilpotent”, “algebraic”

  6. The homogeneous generalized Ricci flow

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  7. Nilpotent Lie algebras obtained by ordered sets and Ricci solitons

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  8. Homogeneous Generalized Ricci flows II

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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 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) 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) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

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.