Paper · Jun 13, 2024

Regularizing property of the twisted conical Kähler-Ricci flow

Jiawei Liu, Shiyu Zhang, Xi Zhang

math.CVarXiv:2406.08778
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Abstract

In this paper, we show the regularity and uniqueness of the twisted conical Kähler-Ricci flow running from a positive closed current with zero Lelong number, which extends the regularizing property of the smooth twisted Kähler-Ricci flow, known as Guedj-Zeriahi's existence theorem and Di Nezza-Lu's uniqueness theorem, to the conical singularity case.

Comments: All comments are welcome! arXiv admin note: substantial text overlap with arXiv:2406.04590

Ricci flowKähler–Ricci flowSingularitiesKähler geometryStability & uniqueness

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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)6Wu 2022 Integral curvature estimates for solutions to Ricci flow with L^p bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)5Liu 2024 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)1Liu 2024 Uniqueness of asymptotically conical Kähler-Ricci flow (Longteng Chen, 2025)7Chen 2025 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)8Zhang 2025 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)2Di Nezza 2026 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)3Di Nezza 2026 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)4Qin 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 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 · Shared terms “zero Lelong”, “closed current”, “Lelong numbers”

  3. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “closed current”, “Lelong numbers”, “positive closed”

  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 · Shared terms “twisted Kähler”, “Nezza”, “Guedj”

  5. 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

  6. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  7. Uniqueness of asymptotically conical Kähler-Ricci flow

    Longteng Chen PDF

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

  8. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang 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.