Paper · Nov 21, 2023

A Hilbert–Mumford criterion for nilsolitons

Yoshinori Hashimoto

math.DGarXiv:2311.12469v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi–Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.

View PDF View HTML arXiv page
Submitted
Nov 21, 2023
Last revised
Nov 18, 2025 (v3)
Subjects
math.DG, math.AG

Comments: v2: 24 pages. Reference added, various minor revisions and corrections. v3: 22 pages. Exposition improved, to appear in Trans. Amer. Math. Soc

Ricci solitonsHomogeneous 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 Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)6Bakhshandeh- 2021 Nilpotent Lie algebras obtained by quivers and Ricci solitons (Fumika Mizoguchi and Hiroshi Tamaru, 2024)2Mizoguchi 2024 Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds (Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso, 2025)7Conti 2025 Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two (Hamid Reza Salimi Moghaddam, 2026)3Moghaddam 2026 Nilpotent Lie algebras obtained by ordered sets and Ricci solitons (Yihao Zheng and Shenglin Zhu, 2026)1Zheng 2026 Solvable Lie algebras obtained by quivers and Einstein metrics (Fumika Mizoguchi, 2026)5Mizoguchi 2026 Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups (Youssef Ayad, 2026)8Ayad 2026 Harmonicity and Existence of Algebraic Generalized Ricci Solitons (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)4Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nilpotent Lie algebras obtained by ordered sets and Ricci solitons

    Yihao Zheng and Shenglin Zhu

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie algebras”, “Lie groups”

  2. Nilpotent Lie algebras obtained by quivers and Ricci solitons

    Fumika Mizoguchi and Hiroshi Tamaru

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie algebras”, “Lie groups”

  3. 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 “Lie groups”, “nilsolitons”, “left invariant”

  4. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “nilpotent Lie”, “Lie algebras”

  5. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie algebras”, “nilpotent Lie”, “admit left”

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

    Rohollah Bakhshandeh-Chamazkoti

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

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

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “nilpotent”, “Lie algebras”

  8. Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

    Youssef Ayad

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie groups”, “algebraic”

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) 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.