Paper · Apr 2, 2017

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

Gregory Edwards

math.DGarXiv:1704.00360
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Abstract

We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point and Gromov-Hausdorff converges to a two dimensional projective orbifold. This gives the first example of the conical Kähler-Ricci flow contracting the cone divisor to a single point. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kähler surfaces in general.

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Submitted
Apr 2, 2017
Subjects
math.DG

Comments: 27 pages, 1 figure

Ricci flowKähler–Ricci flowSingularities(Non)collapsingSurfacesKähler geometryLimit spaces & RCD

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 Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)7Nomura 2016 On a twisted conical Kähler-Ricci flow (Yashan Zhang, 2017)4Zhang 2017 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)6Liu 2019 The Chern-Ricci flow on primary Hopf surfaces (Gregory Edwards, 2019)2Edwards 2019 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)1Liu 2024 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)3Li 2025 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)8Cifarelli 2026 Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I (Yeyun Xu and Linfeng Zhou, 2026)5Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “conical Kähler”, “cone angle”

  2. The Chern-Ricci flow on primary Hopf surfaces

    Gregory Edwards

    Topics (Non)collapsing, Limit spaces & RCD, Kähler geometry · Same author Gregory Edwards · Shared terms “Kähler surfaces”, “Gromov Hausdorff”, “finite time”

  3. On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

    Jiangtao Li

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “Hirzebruch surface”, “finite time”, “orbifold”

  4. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

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

  5. Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

    Yeyun Xu and Linfeng Zhou

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Shared terms “Kähler surfaces”, “finite time”

  6. Stability of the conical Kähler-Ricci flows on Fano manifolds

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “conical Kähler”, “cone angle”

  7. Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “conical Kähler”, “finite time”, “contraction”

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

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al.

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “Kähler surfaces”, “finite time”

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) (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)

In the guidebook

  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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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