Paper · Sep 11, 2018

Kahler-Ricci flow on blowups along submanifolds

Bin Guo

math.DGarXiv:1809.03654
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Abstract

In this short note, we study the behavior of Kaher-Ricci flow on Kahler manifolds which contract divisors to smooth submanifolds. We show that the Kahler potentials are Holder continuous and the flow converges sequentially in Gromov-Hausdorff topology to a compact metric space which is homeomorphic to the base manifold.

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Submitted
Sep 11, 2018
Subjects
math.DG
Ricci flowKähler–Ricci flowSingularitiesKähler geometryLimit spaces & RCD

Its neighbourhood

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 The Kähler-Ricci flow on Fano bundles (Xin Fu and Shijin Zhang, 2016)1Fu 2016 Schauder estimates for equations with cone metrics, I (Bin Guo and Jian Song, 2016)7Guo 2016 Metric contraction of the cone divisor by the conical Kähler-Ricci flow (Gregory Edwards, 2017)8Edwards 2017 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)5Guo 2018 Schauder estimates for equations with cone metrics, II (Bin Guo and Jian Song, 2018)3Guo 2018 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)6Guo 2020 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)2Hallgren 2023 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)4Deruelle 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Kähler-Ricci flow on Fano bundles

    Xin Fu and Shijin Zhang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “sequentially”, “Gromov Hausdorff”

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

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Gromov Hausdorff”, “continuous”

  3. Schauder estimates for equations with cone metrics, II

    Bin Guo and Jian Song

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Bin Guo

  4. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

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

  5. Compactness of Kahler-Ricci solitons on Fano manifolds

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Limit spaces & RCD, Kähler geometry, Singularities · Same author Bin Guo · Shared terms “Hausdorff topology”, “Gromov Hausdorff”

  6. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci flow, Kähler geometry · Same author Bin Guo

  7. Schauder estimates for equations with cone metrics, I

    Bin Guo and Jian Song

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Bin Guo

  8. Metric contraction of the cone divisor by the conical Kähler-Ricci flow

    Gregory Edwards

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Gromov Hausdorff”, “contracts”

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 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) 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.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.