Paper · Jun 30, 2021

Leafwise flat forms on Inoue-Bombieri surfaces

Daniele Angella, Valentino Tosatti

math.DGarXiv:2106.16141v2
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Abstract

We prove that every Gauduchon metric on an Inoue-Bombieri surface admits a strongly leafwise flat form in its -class. Using this result, we deduce uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces. We also show that the convergence is smooth with bounded curvature for initial metrics in the -class of the Tricerri/Vaisman metric.

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Submitted
Jun 30, 2021
Last revised
Apr 26, 2023 (v2)
Subjects
math.DG, math.CV
Journal
J. Funct. Anal. 285 (2023), no.5, Paper No. 110015
DOI
10.1016/j.jfa.2023.110015

Comments: 24 pages. Final version to appear in J. Funct. Anal

Ricci flowGeneralized & G₂ flows

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 Weak Solutions of the Chern-Ricci flow on compact complex surfaces (Xiaolan Nie, 2017)6Nie 2017 On Gauduchon connections with Kähler-like curvature (Daniele Angella, Antonio Otal, Luis Ugarte et al., 2018)4Angella 2018 Geometric formalities along the Chern-Ricci flow (Daniele Angella and Tommaso Sferruzza, 2019)1Angella 2019 The Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)2Tosatti 2021 The behavior of the second Ricci flow on complex parallelizable manifolds (Lucio Bedulli and Luigi Vezzoni, 2024)8Bedulli 2024 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)5Angella 2025 Chern-Ricci flow and t-Gauduchon Ricci-flat condition (Eder M. Correa, Giovane Galindo and Lino Grama, 2025)7Correa 2025 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)3Angella 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric formalities along the Chern-Ricci flow

    Daniele Angella and Tommaso Sferruzza

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Inoue”, “Chern Ricci”

  2. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

    Topics Generalized & G₂ flows · Same author Valentino Tosatti · Shared terms “Chern Ricci”

  3. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “leafwise”, “Chern Ricci”

  4. On Gauduchon connections with Kähler-like curvature

    Daniele Angella, Antonio Otal, Luis Ugarte et al.

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Gauduchon”, “Chern”

  5. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Chern Ricci”

  6. Weak Solutions of the Chern-Ricci flow on compact complex surfaces

    Xiaolan Nie

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”

  7. Chern-Ricci flow and t-Gauduchon Ricci-flat condition

    Eder M. Correa, Giovane Galindo and Lino Grama

    Topics Generalized & G₂ flows · Shared terms “Gauduchon”, “Chern Ricci”

  8. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”

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)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.