Paper · Oct 11, 2018

Expanding solitons to the Hermitian curvature flow on complex Lie groups

Mattia Pujia

math.DGarXiv:1810.05063v2
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Abstract

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.

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Submitted
Oct 11, 2018
Last revised
Mar 5, 2019 (v2)
Subjects
math.DG
Journal
Differential Geom. Appl., Volume 64, 2019, 201-216
DOI
10.1016/j.difgeo.2019.03.001

Comments: 14 pages; section 4 extended; last version, to appear in Differential Geometry and its Applications

Generalized & G₂ flowsRicci solitonsHomogeneous spaces

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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 2016, 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 Laplacian flow of homogeneous G2-structures and its solitons (Jorge Lauret, 2016)5Lauret 2016 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)3Lafuente 2018 Hermitian curvature flow on complex locally homogeneous surfaces (Francesco Pediconi and Mattia Pujia, 2019)6Pediconi 2019 Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups (Mattia Pujia, 2020)1Pujia 2020 Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups (James Stanfield, 2020)4Stanfield 2020 Positive Hermitian Curvature Flow on special linear groups and perfect solitons (James Stanfield, 2021)7Stanfield 2021 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)2Giudice 2025 Generalized Ricci Solitons on Heisenberg and Oscillator Algebras (Vicente Cortés, Shubham Dwivedi and Lorenz Jetter, 2026)8Cortés 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

    Mattia Pujia

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Mattia Pujia · Shared terms “Hermitian curvature”, “algebraic soliton”, “Lie groups”

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

    Ettore Lo Giudice

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”, “algebraic soliton”, “semi algebraic”

  3. Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

    Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Mattia Pujia · Shared terms “Hermitian curvature”, “Lie groups”, “invariant metrics”

  4. Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

    James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “complex Lie”, “Hermitian curvature”, “Lie groups”

  5. Laplacian flow of homogeneous G2-structures and its solitons

    Jorge Lauret

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “algebraic soliton”, “semi algebraic”, “Lie groups”

  6. Hermitian curvature flow on complex locally homogeneous surfaces

    Francesco Pediconi and Mattia Pujia

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Mattia Pujia · Shared terms “Hermitian curvature”

  7. Positive Hermitian Curvature Flow on special linear groups and perfect solitons

    James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”, “complex Lie”, “Lie groups”

  8. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “algebraic soliton”, “semi algebraic”, “Lie groups”

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.

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)

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.