Paper · Feb 13, 2022

Tangent Flows of Kähler Metric Flows

Max Hallgren, Wangjian Jian

math.DGarXiv:2202.06185
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Abstract

We improve the description of -limits of noncollapsed Ricci flows in the Kähler setting. In particular, the singular strata of such metric flows satisfy . We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial one-parameter action by isometries, which is locally free on the cone link in the static case. The main results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.

Comments: 35 pages. All comments are welcome!

Ricci flowSingularities(Non)collapsingHeat kernelKähler geometry

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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 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 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)1Hallgren 2023 Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Ronan J. Conlon, Max Hallgren and Zilu Ma, 2025)6Conlon 2025 An ε-Regularity Theorem for Non-collapsed Ricci Flow (Harry Fluck and Max Hallgren, 2025)3Fluck 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)4Hallgren 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)7Chen 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)8Jian 2026 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)2Cifarelli 2026 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)5Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler geometry, Singularities · Same authors Max Hallgren, Wangjian Jian

  2. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al. PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “flow satisfies”, “tangent flow”

  3. An -Regularity Theorem for Non-collapsed Ricci Flow

    Harry Fluck and Max Hallgren PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “parabolic regularizations”

  4. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang PDF

    Topics (Non)collapsing, Kähler geometry · Same author Max Hallgren · Shared terms “almost selfsimilar”, “flow admits”, “noncollapsed”

  5. Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

    Lei Zhang and Zhenlei Zhang PDF

    Topics Heat kernel, (Non)collapsing, Kähler geometry · Shared terms “conjugate heat”

  6. Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Ronan J. Conlon, Max Hallgren and Zilu Ma PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren

  7. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “tangent flow”, “analogous”, “noncollapsed”

  8. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Wangjian Jian · Shared terms “tangent flow”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.