Paper · Jun 21, 2017

Hermitian curvature flow on complex homogeneous manifolds

Yury Ustinovskiy

math.DGarXiv:1706.07023
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Abstract

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on -homogeneous manifolds, where is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or solvable complex Lie group equipped with an induced metric we investigate the blow-up behavior of the HCF.

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Submitted
Jun 21, 2017
Subjects
math.DG, math.CV

Comments: 14 pages

Generalized & G₂ flowsEinstein metricsSingularitiesSmale conjecture & diffeomorphismsHomogeneous 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 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 The Hermitian curvature flow on manifolds with non-negative Griffiths curvature (Yury Ustinovskiy, 2016)4Ustinovskiy 2016 Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow (Yury Ustinovskiy, 2017)3Ustinovskiy 2017 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)7Lafuente 2018 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)2Pujia 2018 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)5Panelli 2019 On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature (Yury Ustinovskiy, 2019)1Ustinovskiy 2019 Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups (James Stanfield, 2020)6Stanfield 2020 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)8Giudice 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “complex Lie”

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

    Mattia Pujia

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

  3. Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “Lie”

  4. The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “Hermitian curvature”, “HCF”

  5. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

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

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

    James Stanfield

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

  7. 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 · Shared terms “Hermitian curvature”, “HCF”, “Lie groups”

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

    Ettore Lo Giudice

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

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Smale conjecture & diffeomorphisms Once Ricci flow through singularities is unique and continuous, it can deform whole families of metrics at once. Start with: A Proof of the Smale Conjecture, Diff(S 3 ) ≃O(4) (Hatcher, 1983)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.