Paper · Mar 21, 2025

Hermitian metrics on complex non-Kähler manifolds

Daniele Angella

math.CVarXiv:2503.16936v2
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Abstract

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the Kähler-Einstein condition to the non-Kähler setting, and the convergence of the Chern-Ricci flow on compact complex surfaces.

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Submitted
Mar 21, 2025
Last revised
May 16, 2025 (v2)
Subjects
math.CV, math.DG

Comments: Dedicated to Professor Paul Gauduchon, on his 80th birthday. This survey expands the topic of a seminar given by the author at the Second International Conference on Differential Geometry in Fes in October 2024

Ricci flowGeneralized & G₂ flowsEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stability

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)2Tosatti 2021 Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds (Eder M. Correa, 2022)8Correa 2022 The behavior of the second Ricci flow on complex parallelizable manifolds (Lucio Bedulli and Luigi Vezzoni, 2024)3Bedulli 2024 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)1Gallo 2026 Scalar Curvature, Volumes and the Bergman Kernel (Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang, 2026)6Chen 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)4Angella 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)5Huang 2026 Continuation and convergence for the higher dimensional conformal Chern–Calabi flow (Hongyu Chen, Li Ma and Li Sheng, 2026)7Chen 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 Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

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

  2. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

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

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

    Lucio Bedulli and Luigi Vezzoni

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

  4. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

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

  5. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng

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

  6. Scalar Curvature, Volumes and the Bergman Kernel

    Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang

    Topics Kähler–Einstein & K-stability, Einstein metrics, Scalar curvature · Shared terms “compact complex”, “Chern scalar”, “Hermitian”

  7. Continuation and convergence for the higher dimensional conformal Chern–Calabi flow

    Hongyu Chen, Li Ma and Li ShengWider flows

    Topics Kähler–Einstein & K-stability, Einstein metrics, Scalar curvature · Shared terms “Chern scalar”, “constant Chern”, “Hermitian”

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

    Eder M. Correa

    Topics Generalized & G₂ flows, Einstein metrics, Scalar curvature · Shared terms “Hermitian metrics”, “Chern”, “Hermitian manifolds”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  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.