Paper · Mar 18, 2019

Stability of the conical Kähler-Ricci flows on Fano manifolds

Jiawei Liu, Xi Zhang

math.DGarXiv:1903.07528v2
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Abstract

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle along the divisor, then for any sufficiently close to , the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.

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Submitted
Mar 18, 2019
Last revised
Apr 16, 2019 (v2)
Subjects
math.DG, math.AP

Comments: 51 pages, we improve the writing and add a remark

Ricci flowKähler–Ricci flowEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability

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)1Liu 2016 A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces (Yashan Zhang, 2016)6Zhang 2016 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)3Liu 2017 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)5Guo 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)8Wang 2020 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)7Jian 2023 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)2Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)4Liu 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, Kähler geometry, Stability & uniqueness · Same authors Jiawei Liu, Xi Zhang · Shared terms “conical Kähler”, “Fano manifolds”

  2. 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 · Same authors Jiawei Liu, Xi Zhang · Shared terms “conical Kähler”, “cone angle”, “flow converges”

  3. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same authors Jiawei Liu, Xi Zhang · Shared terms “conical Kähler”, “flow converges”

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

    Jiawei Liu, Shiyu Zhang and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Stability & uniqueness · Same authors Jiawei Liu, Xi Zhang · Shared terms “conical Kähler”

  5. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Fano manifolds”

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

    Yashan Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “conical Kähler”

  7. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Fano manifolds”

  8. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Fano manifolds”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 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.