Paper · Jul 27, 2021

The Chern-Ricci flow

Valentino Tosatti, Ben Weinkove

math.DGarXiv:2107.12928v3Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 33 (2022), no.1, 73-107
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Abstract

We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.

Comments: 30 pages; final version to appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl

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 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 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)8Dang 2023 The behavior of the second Ricci flow on complex parallelizable manifolds (Lucio Bedulli and Luigi Vezzoni, 2024)1Bedulli 2024 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)5Wondo 2024 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)3Angella 2025 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)2Gallo 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)7Angella 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)4Huang 2026 Chern holonomy under the first and second Chern–Ricci flows (Yuan Liao, 2026)6Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni PDF

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

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

    Marco Gallo and Luigi Vezzoni PDF

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

  3. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella PDF

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

  4. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng PDF

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

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

    Hosea Wondo PDF

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

  6. Chern holonomy under the first and second Chern–Ricci flows

    Yuan Liao PDF

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

  7. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa PDF

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

  8. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang PDF

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

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.