Paper · Oct 11, 2023

Geometric regularity of blow-up limits of the Kähler-Ricci flow

Max Hallgren, Wangjian Jian, Jian Song, Gang Tian

math.DGarXiv:2310.08610
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Abstract

We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov- distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than .

Comments: All comments welcome. arXiv admin note: text overlap with arXiv:2310.07945

Ricci flowKähler–Ricci flowSingularitiesKähler geometryLimit spaces & RCD

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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 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 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)3Tian 2022 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)7Miao 2022 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)2Jian 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)5Jian 2023 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)6Jian 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)8Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)4Jian 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)1Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “Gromov Hausdorff”, “limiting”

  2. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same authors Wangjian Jian, Jian Song, Gang Tian · Shared terms “Gromov Hausdorff”

  3. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “Gromov Hausdorff”

  4. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song

  5. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song, Gang Tian

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

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song · Shared terms “limiting”

  7. A Note On Kähler-Ricci Flow on Fano Threefolds

    Minghao Miao and Gang Tian PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “Gromov Hausdorff”

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

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song · Shared terms “limiting”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) 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. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 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.