Paper · Dec 11, 2018

Second Ricci flow on noncompact Hermitian manifolds

Man-Chun Lee

math.DGarXiv:1812.04610v2
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Abstract

In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.

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Submitted
Dec 11, 2018
Last revised
Dec 2, 2019 (v2)
Subjects
math.DG

Comments: 28 pages

Ricci flowEinstein metricsKähler geometryBoundaries & non-compactHarmonic maps & DeTurck trickKähler–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)2Lee 2017 On the uniqueness of Ricci flow (Man-Chun Lee, 2017)6Lee 2017 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)4Huang 2018 Hermitian manifolds with quasi-negative curvature (Man-Chun Lee, 2018)5Lee 2018 Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics (Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam, 2019)1Huang 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)3Lee 2019 Uniqueness of Ricci flow with scaling invariant estimates (Man-Chun Lee, 2025)7Lee 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)8Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics

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

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

  2. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Kähler geometry · Same author Man-Chun Lee · Shared terms “complete noncompact”, “short time”, “time existence”

  3. 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

  4. 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, Kähler geometry · Same author Man-Chun Lee

  5. Hermitian manifolds with quasi-negative curvature

    Man-Chun Lee

    Same author Man-Chun Lee · Shared terms “Hermitian manifolds”

  6. On the uniqueness of Ricci flow

    Man-Chun Lee

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “complete noncompact”

  7. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Same author Man-Chun Lee

  8. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

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

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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  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.