Paper · Oct 1, 2026

Solvable Algebraic Generalized Ricci Solitons in low dimensions

Lorenzo Mucciante

math.DGarXiv:2610.01608
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Abstract

We prove new structural results for algebraic generalized Ricci solitons on solvable Lie algebras and we discuss the existence of examples in low dimensions. We establish that all non-trivial solvable algebraic generalized Ricci solitons must be expanding and give necessary conditions for the existence of such solitons on one-dimensional extensions of abelian Lie algebras. Finally, we classify algebraic generalized Ricci solitons on three-dimensional Lie algebras, and algebraic generalized Ricci solitons on four-dimensional solvable unimodular Lie algebras, up to isometry and scaling.

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Submitted
Oct 1, 2026
Subjects
math.DG
Generalized & G₂ flowsRicci solitons3-manifolds4-manifoldsHomogeneous spaces

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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

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “algebraic generalized”, “solvable”, “Lie algebras”

  2. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “algebraic generalized”, “Lie algebras”

  3. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “dimensional Lie”, “Lie algebras”, “algebraic”

  4. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “low dimensions”, “Lie”

  5. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”, “Lie”

  6. Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, 3-manifolds, Homogeneous spaces

  7. Four-dimensional Lorentzian algebraic Ricci solitons

    Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “algebraic”, “dimensional Lie”, “four dimensional”

  8. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim

    Topics 4-manifolds, 3-manifolds, Homogeneous spaces · Shared terms “four dimensional”, “Lie”, “isometry”

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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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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