Paper · May 15, 2017

Cusp Kähler-Ricci flow on compact Kähler manifold

Jiawei Liu, Xi Zhang

math.DGarXiv:1705.05129
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Abstract

In this paper, by limiting twisted conical Kähler-Ricci flows, we prove the long-time existence and uniqueness of cusp Kähler-Ricci flow on compact Kähler manifold which carries a smooth hypersurface such that the twisted canonical bundle is ample. Furthermore, we prove that this flow converge to a unique cusp Kähler-Einstein metric.

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Submitted
May 15, 2017
Subjects
math.DG, math.AP
Ricci flowKähler–Ricci flowEinstein metricsKähler geometryStability & uniquenessLong-time existence & convergenceKähler–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)4Liu 2016 A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces (Yashan Zhang, 2016)5Zhang 2016 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)8Huang 2018 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)1Liu 2019 Convergence of the weak Kähler-Ricci Flow on manifolds of general type (Tat Dat Tô, 2019)6Tô 2019 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)3Liu 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)7Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the conical Kähler-Ricci flows on Fano manifolds

    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”

  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 “twisted conical”, “conical Kähler”, “flow converges”

  3. 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 “twisted conical”, “conical Kähler”

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

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Jiawei Liu, Xi Zhang · Shared terms “conical Kähler”

  5. 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”, “twisted canonical”

  6. Convergence of the weak Kähler-Ricci Flow on manifolds of general type

    Tat Dat Tô

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

  7. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

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

  8. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics

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.