Paper · Apr 6, 2017

Smooth approximation of the modified conical Kähler-Ricci flow

Ryosuke Takahashi

math.DGarXiv:1704.01879v2
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Abstract

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

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Submitted
Apr 6, 2017
Last revised
Apr 11, 2017 (v2)
Subjects
math.DG

Comments: 24 pages, typos are fixed, Section 1, Theorem 1.1 and References are updated

Ricci flowKähler–Ricci flowKähler geometryLong-time existence & convergence

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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 The conical Kähler-Ricci flow with weak initial data on Fano manifold (Jiawei Liu and Xi Zhang, 2016)1Liu 2016 Bergman iteration and C^(∞)-convergence towards Kähler-Ricci flow (Ryosuke Takahashi, 2016)7Takahashi 2016 Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)4Nomura 2016 On a twisted conical Kähler-Ricci flow (Yashan Zhang, 2017)2Zhang 2017 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)8Liu 2019 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)5Wu 2022 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)3Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)6Liu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The conical Kähler-Ricci flow with weak initial data on Fano manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Shared terms “conical Kähler”

  2. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “conical Kähler”, “smooth approximation”, “holomorphic”

  3. 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 · Shared terms “conical Kähler”

  4. 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 · Shared terms “conical Kähler”

  5. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “modified”

  6. Regularizing property of the twisted conical Kähler-Ricci flow

    Jiawei Liu, Shiyu Zhang and Xi Zhang

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

  7. Bergman iteration and -convergence towards Kähler-Ricci flow

    Ryosuke Takahashi

    Topics Kähler–Ricci flow, Kähler geometry · Same author Ryosuke Takahashi · Shared terms “sequence”

  8. 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”

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.