Paper · Jun 11, 2019

The K"ahler-Ricci flow with Log Canonical Singularities

Albert Chau, Huabin Ge, Ka-Fai Li, Liangming Shen

math.DGarXiv:1906.04343v3
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Abstract

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Finally we also construct K"ahler-Ricci flow solutions performing divisorial contractions and flips with log canonical singularities.

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Submitted
Jun 11, 2019
Last revised
Jul 12, 2022 (v3)
Subjects
math.DG

Comments: Correction to Lemma 3.1 and Theorem 3.13; detail added to Lemma 6.1 and proof; notation simplified throughout; To appear in International Mathematics Research Notices

Ricci flowEinstein metricsSingularitiesAnalytic minimal model program

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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 Kähler-Ricci flow of cusp singularities on quasi projective varieties (Albert Chau, Ka-Fai Li and Liangming Shen, 2017)1Chau 2017 The Kähler-Ricci flow on pseudoconvex domains (Huabin Ge, Aijin Lin and Liangming Shen, 2018)2Ge 2018 Existence of Kahler-Ricci solitons on smoothable Q-Fano varities (Yan Li, 2019)5Li 2019 On the Hölder estimate of Kähler-Ricci flow (Jianchun Chu and Man-Chun Lee, 2021)3Chu 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)7Fung 2021 Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2022)6Chang 2022 Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds (Shu-Cheng Chang, Chien Lin and Chin-Tung Wu, 2022)4Chang 2022 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. Kähler-Ricci flow of cusp singularities on quasi projective varieties

    Albert Chau, Ka-Fai Li and Liangming Shen

    Topics Singularities · Same authors Albert Chau, Ka-Fai Li, Liangming Shen · Shared terms “projective varieties”

  2. The Kähler-Ricci flow on pseudoconvex domains

    Huabin Ge, Aijin Lin and Liangming Shen

    Topics Einstein metrics · Same authors Huabin Ge, Liangming Shen

  3. On the Hölder estimate of Kähler-Ricci flow

    Jianchun Chu and Man-Chun Lee

    Topics Analytic minimal model program · Shared terms “ahler Ricci”

  4. Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

    Shu-Cheng Chang, Chien Lin and Chin-Tung Wu

    Topics Analytic minimal model program, Singularities · Shared terms “divisorial contractions”, “Song Tian”

  5. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    Yan Li

    Shared terms “ahler Ricci”

  6. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al.

    Topics Analytic minimal model program, Einstein metrics, Singularities · Shared terms “canonical model”

  7. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

    Topics Analytic minimal model program, Einstein metrics

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Analytic minimal model program, Einstein metrics · Shared terms “Song Tian”, “canonical model”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Analytic minimal model program On a projective variety the Kähler–Ricci flow runs until its Kähler class stops being Kähler, a time read off from cohomology. Start with: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type (Tsuji, 1988)

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    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.