Paper · Feb 17, 2020

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

Mattia Pujia

math.DGarXiv:2002.07210v2
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Abstract

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger- Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

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Submitted
Feb 17, 2020
Last revised
Sep 21, 2020 (v2)
Subjects
math.DG

Comments: 9 pages. Some minor changes. To appear on manuscripta math

Generalized & G₂ flowsHomogeneous spacesStability & uniquenessLong-time existence & convergence

Its neighbourhood

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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)7Lauret 2016 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)4Lafuente 2018 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)3Pujia 2018 Hermitian curvature flow on complex locally homogeneous surfaces (Francesco Pediconi and Mattia Pujia, 2019)6Pediconi 2019 Generalized Ricci flow on nilpotent Lie groups (Fabio Paradiso, 2020)8Paradiso 2020 Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups (James Stanfield, 2020)2Stanfield 2020 Positive Hermitian Curvature Flow on special linear groups and perfect solitons (James Stanfield, 2021)5Stanfield 2021 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)1Giudice 2025
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 2-step nilpotent Lie groups

    Ettore Lo Giudice

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “positive Hermitian”, “step nilpotent”, “Hermitian curvature”

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

    James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “positive Hermitian”, “Hermitian curvature”, “Lie groups”

  3. Expanding solitons to the Hermitian curvature flow on complex Lie groups

    Mattia Pujia

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

  4. 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, Long-time existence & convergence · Same author Mattia Pujia · Shared terms “Hermitian curvature”, “Lie groups”, “invariant metrics”

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

    James Stanfield

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

  6. Hermitian curvature flow on complex locally homogeneous surfaces

    Francesco Pediconi and Mattia Pujia

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Same author Mattia Pujia · Shared terms “Hermitian curvature”, “characterize”, “Gromov”

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

    Jorge Lauret

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “algebraic soliton”, “Lie groups”, “nilpotent Lie”

  8. Generalized Ricci flow on nilpotent Lie groups

    Fabio Paradiso

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “nilpotent Lie”, “Lie groups”, “left invariant”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.