Paper · Jul 11, 2016

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

Ryosuke Nomura

math.DGarXiv:1607.03004v2
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Abstract

We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow , which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time . We prove that the scalar curvature of is bounded from above by under the existence of a contraction associated to the limiting cohomology class . This generalizes Z. Zhang's work to the conic case.

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Submitted
Jul 11, 2016
Last revised
Feb 15, 2018 (v2)
Subjects
math.DG, math.CV
Journal
Differential Geometry and its Applications, Volume 58, June 2018, Pages 1-16
DOI
10.1016/j.difgeo.2017.12.001

Comments: 16pages, minor changes

Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvature

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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same author Ryosuke Nomura · Shared terms “curvature along”

  2. Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

    Alexander Bednarek

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “limiting cohomology”, “finite time”

  3. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

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

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

    Ryosuke Takahashi

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

  5. 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”

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

    Jiawei Liu, Shiyu Zhang and Xi Zhang

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

  7. 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, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  8. 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”

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.