Paper · Jun 7, 2024

Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

Jiawei Liu, Xi Zhang

math.DGarXiv:2406.04590
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Abstract

In this paper, we study the limit behavior of the conical Kähler-Ricci flow as its cone angle tends to zero. More precisely, we prove that as the cone angle tends to zero, the conical Kähler-Ricci flow converges to a unique Kähler-Ricci flow, which is smooth outside the divisor and admits cusp singularity along the divisor.

Comments: All comments are welcome!

Ricci flowKähler–Ricci flowSingularitiesKähler geometry

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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 2021, 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 modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)3Wu 2022 Kähler-Ricci flow and conformal submersion (Hoan Nguyen, 2023)5Nguyen 2023 Integral curvature estimates for solutions to Ricci flow with L^p bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)6Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)1Liu 2024 Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities (Alexander Bednarek, 2024)8Bednarek 2024 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)2Di Nezza 2026 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)4Qin 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)7Van Why 2026
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 PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Jiawei Liu, Xi Zhang · Shared terms “twisted conical”, “conical Kähler”

  2. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  3. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  4. Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

    Xiangsen Qin PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  5. Kähler-Ricci flow and conformal submersion

    Hoan Nguyen PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  6. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jiawei Liu

  7. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why PDF

    Topics Kähler–Ricci flow, Kähler geometry

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

    Alexander Bednarek PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

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 usually cannot run forever: curvature blows up in finite time, 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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… 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.