Paper · May 30, 2019

The Chern-Ricci flow on primary Hopf surfaces

Gregory Edwards

math.DGarXiv:1905.13114
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Abstract

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern-Ricci flow. We use a construction of Gauduchon-Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern-Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round . Uniform estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

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Submitted
May 30, 2019
Subjects
math.DG
Ricci flowGeneralized & G₂ flowsSingularities(Non)collapsingKähler geometryLimit spaces & RCD

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 Metric contraction of the cone divisor by the conical Kähler-Ricci flow (Gregory Edwards, 2017)2Edwards 2017 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)6Tosatti 2021 Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds (Eder M. Correa, 2022)5Correa 2022 Chern-Ricci flow and t-Gauduchon Ricci-flat condition (Eder M. Correa, Giovane Galindo and Lino Grama, 2025)3Correa 2025 Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type (Haoyuan Sun, 2026)8Sun 2026 Convergence of the Chern-Ricci flow on complex minimal surfaces of general type (Haoyuan Sun, 2026)4Sun 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)7Angella 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 · Shared terms “Chern Ricci”, “Hopf”, “VII”

  2. Metric contraction of the cone divisor by the conical Kähler-Ricci flow

    Gregory Edwards

    Topics (Non)collapsing, Limit spaces & RCD, Kähler geometry · Same author Gregory Edwards · Shared terms “Kähler surfaces”, “Gromov Hausdorff”, “finite time”

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

    Eder M. Correa, Giovane Galindo and Lino Grama

    Topics Generalized & G₂ flows, Limit spaces & RCD, Kähler geometry · Shared terms “Chern Ricci”, “Gauduchon”, “Hopf”

  4. Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

    Haoyuan Sun

    Topics Generalized & G₂ flows, (Non)collapsing, Limit spaces & RCD · Shared terms “Chern Ricci”, “Gromov Hausdorff”

  5. Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds

    Eder M. Correa

    Topics Generalized & G₂ flows, (Non)collapsing, Limit spaces & RCD · Shared terms “Hopf”, “Chern Ricci”, “Hausdorff limit”

  6. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

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

  7. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

    Topics Generalized & G₂ flows, Limit spaces & RCD · Shared terms “Chern Ricci”, “Gromov Hausdorff”

  8. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    Haoyuan Sun

    Topics Generalized & G₂ flows, (Non)collapsing, Limit spaces & RCD · Shared terms “Chern Ricci”, “Hausdorff limit”, “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.

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.