Paper · Feb 11, 2019

Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics

Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam

math.DGarXiv:1902.04017v2
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Abstract

In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as . These results can be viewed as generalization of an existence result by Giesen and Topping for surfaces of hyperbolic type of Ricci flow to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete noncompact Hermitian manifolds, which generalizes the work of Lott-Zhang and Tosatti-Weinkove to complete noncompact Hermitian manifolds with possibly unbounded curvature.

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Submitted
Feb 11, 2019
Last revised
Aug 15, 2019 (v2)
Subjects
math.DG

Comments: 40 pages

Ricci flowGeneralized & G₂ flowsEinstein metricsKähler geometryBoundaries & non-compactLong-time existence & convergenceKähler–Einstein & K-stability

Its neighbourhood

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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 On existence and curvature estimates of Ricci flow (Man-Chun Lee and Luen-Fai Tam, 2017)8Lee 2017 Chern-Ricci flows on noncompact complex manifolds (Man-Chun Lee and Luen-Fai Tam, 2017)4Lee 2017 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)3Huang 2018 Second Ricci flow on noncompact Hermitian manifolds (Man-Chun Lee, 2018)2Lee 2018 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)5Lee 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)7Lee 2019 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)6Angella 2025 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)1Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng

    Topics Generalized & G₂ flows, Boundaries & non-compact, Kähler geometry · Same author Shaochuang Huang · Shared terms “complete Chern”, “instantaneously complete”, “Chern Ricci”

  2. Second Ricci flow on noncompact Hermitian manifolds

    Man-Chun Lee

    Topics Kähler–Einstein & K-stability, Einstein metrics, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “noncompact Hermitian”, “Hermitian manifolds”, “complete noncompact”

  3. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Kähler–Einstein & K-stability, Einstein metrics, Long-time existence & convergence · Same authors Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam · Shared terms “possibly unbounded”, “unbounded curvature”

  4. Chern-Ricci flows on noncompact complex manifolds

    Man-Chun Lee and Luen-Fai Tam

    Topics Generalized & G₂ flows, Boundaries & non-compact, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “Chern Ricci”, “complex manifold”, “Lott Zhang”

  5. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Long-time existence & convergence, Boundaries & non-compact, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “complete noncompact”, “complex manifold”, “Topping”

  6. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

    Topics Generalized & G₂ flows, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Chern Ricci”, “Hermitian manifolds”

  7. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same author Man-Chun Lee

  8. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Boundaries & non-compact, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “complete noncompact”, “Topping”

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 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.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.