Paper · Jun 29, 2018

Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

Ramiro A. Lafuente, Mattia Pujia, Luigi Vezzoni

math.DGarXiv:1807.00059v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation . The solution always exist for all positive times, and converges as in Cheeger-Gromov sense to a non-flat left-invariant soliton . Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures.

View PDF View HTML arXiv page
Submitted
Jun 29, 2018
Last revised
Dec 19, 2019 (v2)
Subjects
math.DG
Journal
Trans. Amer. Math. Soc., Volume 373 (6), 2020, 3967-3993
DOI
10.1090/tran/8068

Comments: 25 pages. Revised version. To appear in TAMS

Ricci flowGeneralized & G₂ flowsKähler geometryHomogeneous spacesLong-time existence & convergence

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 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 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)2Pujia 2018 Hermitian curvature flow on complex locally homogeneous surfaces (Francesco Pediconi and Mattia Pujia, 2019)5Pediconi 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 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)8Fusi 2024 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)3Giudice 2025 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)7Gallo 2026 Homogeneous Generalized Ricci flows II (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2026)6Fusi 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, Long-time existence & convergence · Same author Mattia Pujia · Shared terms “Hermitian curvature”, “Lie groups”, “invariant metrics”

  2. 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”, “Lie groups”, “invariant metrics”

  3. 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 “Hermitian curvature”, “Lie groups”, “left invariant”

  4. 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 “Hermitian curvature”, “abelian complex”, “Lie groups”

  5. 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”, “Gromov”

  6. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Ramiro A. Lafuente · Shared terms “unimodular Lie”, “Lie groups”, “left invariant”

  7. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

    Topics Generalized & G₂ flows, Kähler geometry · Same author Luigi Vezzoni · Shared terms “Hermitian metrics”, “Hermitian curvature”

  8. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Same author Ramiro A. Lafuente · Shared terms “Lie groups”, “pluriclosed flow”, “quotient”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.