Paper · Nov 29, 2017

On a twisted conical Kähler-Ricci flow

Yashan Zhang

math.DGarXiv:1711.10756v3
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Abstract

In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical Kähler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical Kähler metric with possibly unbounded scalar curvature, the conical Kähler-Ricci flow will instantly have bounded scalar curvature for , and the bound is of the form . Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted Kähler-Ricci flow and its convergence.

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Submitted
Nov 29, 2017
Last revised
Jun 10, 2018 (v3)
Subjects
math.DG

Comments: V3, small changes; accepted by Annals of Global Analysis and Geometry

Ricci flowKähler–Ricci flowKähler geometryScalar curvatureLimit 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 conical Kähler-Ricci flow with weak initial data on Fano manifold (Jiawei Liu and Xi Zhang, 2016)7Liu 2016 Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)5Nomura 2016 A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces (Yashan Zhang, 2016)3Zhang 2016 Smooth approximation of the modified conical Kähler-Ricci flow (Ryosuke Takahashi, 2017)2Takahashi 2017 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)6Liu 2017 Calabi Symmetry and the Continuity Method (Hosea Wondo, 2022)8Wondo 2022 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)4Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)1Liu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Regularizing property of the twisted conical Kähler-Ricci flow

    Jiawei Liu, Shiyu Zhang and Xi Zhang

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

  2. Smooth approximation of the modified conical Kähler-Ricci flow

    Ryosuke Takahashi

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

  3. A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “conical Kähler”

  4. 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 “twisted conical”, “conical Kähler”

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

    Ryosuke Nomura

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

  6. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

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

  7. The conical Kähler-Ricci flow with weak initial data on Fano manifold

    Jiawei Liu and Xi Zhang

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

  8. Calabi Symmetry and the Continuity Method

    Hosea Wondo

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

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.