Paper · Mar 18, 2017

An almost complex Chern-Ricci flow

Tao Zheng

math.DGarXiv:1703.06295v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the evolution of an almost Hermitian metric by the part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give some convergence results. As an example, we study this flow on the (locally) homogeneous manifolds in more detail.

View PDF View HTML arXiv page
Submitted
Mar 18, 2017
Last revised
Jul 16, 2017 (v3)
Subjects
math.DG
Journal
The Journal of Geometric Analysis \textbf{28} (2018), no. 3, 2129-2165
DOI
10.1007/s12220-017-9898-9.

Comments: 28pages, the final version accepted by The Journal of Geometric Analysis

Ricci flowKähler–Ricci flowGeneralized & G₂ flowsKähler geometryHomogeneous 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 Weak Solutions of the Chern-Ricci flow on compact complex surfaces (Xiaolan Nie, 2017)7Nie 2017 Chern-Ricci flows on noncompact complex manifolds (Man-Chun Lee and Luen-Fai Tam, 2017)3Lee 2017 Symplectic curvature flow revisited (Casey Lynn Kelleher, 2018)8Kelleher 2018 Geometric formalities along the Chern-Ricci flow (Daniele Angella and Tommaso Sferruzza, 2019)4Angella 2019 The Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)1Tosatti 2021 The behavior of the second Ricci flow on complex parallelizable manifolds (Lucio Bedulli and Luigi Vezzoni, 2024)6Bedulli 2024 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)2Wondo 2024 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)5Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

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

  2. Singularity Types for Long-Time Chern-Ricci Flow

    Hosea Wondo

    Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Shared terms “Chern Ricci”, “initial metric”

  3. Chern-Ricci flows on noncompact complex manifolds

    Man-Chun Lee and Luen-Fai Tam

    Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Shared terms “Chern Ricci”, “complex manifold”, “initial metric”

  4. Geometric formalities along the Chern-Ricci flow

    Daniele Angella and Tommaso Sferruzza

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

  5. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “complex structure”, “complex manifold”

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

    Lucio Bedulli and Luigi Vezzoni

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Ricci form”, “complex manifold”

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

    Xiaolan Nie

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

  8. Symplectic curvature flow revisited

    Casey Lynn Kelleher

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “almost complex”, “Chern”, “complex structure”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

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

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

  3. 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.