Paper · Jun 27, 2019

Hermitian curvature flow on complex locally homogeneous surfaces

Francesco Pediconi, Mattia Pujia

math.DGarXiv:1906.11676v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We study the Hermitian curvature flow of locally homogeneous non-Kähler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-Kähler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry.

View PDF View HTML arXiv page
Submitted
Jun 27, 2019
Last revised
Jun 30, 2020 (v2)
Subjects
math.DG, math.CV
DOI
10.1007/s10231-020-01015-z

Comments: Some minor changes. A new appendix containing explicit tensor components. To appear on Ann. Mat. Pura Appl

Generalized & G₂ flowsSingularitiesKähler geometryLimit spaces & RCDHomogeneous 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 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)1Lafuente 2018 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)3Pujia 2018 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)4Panelli 2019 Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups (Mattia Pujia, 2020)2Pujia 2020 Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups (James Stanfield, 2020)7Stanfield 2020 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)8Pediconi 2021 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)6Giudice 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)5Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”, “Gromov”

  2. 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”, “characterize”, “Gromov”

  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”

  4. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

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

  5. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Limit spaces & RCD, Homogeneous spaces, Long-time existence & convergence · Same author Francesco Pediconi · Shared terms “time behavior”

  6. 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”, “time behavior”, “Gromov”

  7. 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”, “Gromov”

  8. Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

    Francesco Pediconi and Sammy Sbiti

    Topics Limit spaces & RCD, Homogeneous spaces · Same author Francesco Pediconi · Shared terms “Gromov Hausdorff”

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) 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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.