Paper · Aug 28, 2023

Derivative estimates of pluriclosed flow

Yanan Ye

math.DGarXiv:2308.14600v3
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As on arXiv

Abstract

We provide a derivative estimate for the pluriclosed flow, controlling higher order derivatives of Chern curvature and torsion using the Chern curvature. Moreover, we derive an estimate for torsion tensor using Chern Ricci curvature in dimension two. And in the Hermitian-symplectic case, we find a monotonic quantity and use it to prove that all Hermitian-symplectic solitons are Kähler Ricci solitons.

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Submitted
Aug 28, 2023
Last revised
Nov 13, 2023 (v3)
Subjects
math.DG
Generalized & G₂ flowsRicci solitonsKähler–Ricci solitonsKähler geometry

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 2021, 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 Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)5Tosatti 2021 Generalized almost-Kähler-Ricci solitons (Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi, 2024)7Albanese 2024 On the shrinking solitons of generalized Ricci flow (Xilun Li and Yanan Ye, 2024)1Li 2024 Chern-Ricci flow and t-Gauduchon Ricci-flat condition (Eder M. Correa, Giovane Galindo and Lino Grama, 2025)4Correa 2025 Dynamical stability of Pluriclosed and Generalized Ricci solitons (Kuan-Hui Lee, 2025)3Lee 2025 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)8Apostolov 2025 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)2Streets 2025 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)6Gallo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the shrinking solitons of generalized Ricci flow

    Xilun Li and Yanan Ye

    Topics Generalized & G₂ flows, Kähler–Ricci solitons, Kähler geometry · Same author Yanan Ye · Shared terms “pluriclosed”

  2. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “pluriclosed flow”

  3. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

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

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

    Eder M. Correa, Giovane Galindo and Lino Grama

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “pluriclosed flow”, “Chern Ricci”, “Hermitian”

  5. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

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

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

    Marco Gallo and Luigi Vezzoni

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Hermitian”, “Chern Ricci”

  7. Generalized almost-Kähler-Ricci solitons

    Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “symplectic”, “Chern”

  8. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “symplectic”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.