Paper · Nov 26, 2019

Some refinements of the partial estimate

Kewei Zhang

math.DGarXiv:1911.11328
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Abstract

Relying on the recent work of Liu-Székelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kähler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial estimate along the (generalized) Kähler-Ricci flow on Fano manifolds.

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Submitted
Nov 26, 2019
Subjects
math.DG
Journal
Analysis & PDE 14 (2021) 2307-2326
DOI
10.2140/apde.2021.14.2307

Comments: Comments are welcome!

Ricci flowKähler–Ricci flowKähler geometry

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.

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)3Liu 2016 Remarks of weak-compactness along Kahler Ricci flow (Xiuxiong Chen and Bing Wang, 2016)8Chen 2016 Bergman iteration and C^(∞)-convergence towards Kähler-Ricci flow (Ryosuke Takahashi, 2016)4Takahashi 2016 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)7Guo 2020 Tian's partial C⁰-estimate implies Hamilton-Tian's conjecture (Feng Wang and Xiaohua Zhu, 2020)1Wang 2020 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)5Jian 2023 Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2024)6Hisamoto 2024 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)2Zhang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Tian's partial -estimate implies Hamilton-Tian's conjecture

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Liu Székelyhidi”, “Fano manifolds”

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

    Junsheng Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “lower bound”, “Fano”

  3. 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 “Fano manifolds”

  4. Bergman iteration and -convergence towards Kähler-Ricci flow

    Ryosuke Takahashi

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

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

    Wangjian Jian and Yalong Shi

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Fano manifolds”, “estimate along”

  6. Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

    Tomoyuki Hisamoto

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

  7. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

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

  8. Remarks of weak-compactness along Kahler Ricci flow

    Xiuxiong Chen and Bing Wang

    Topics Kähler–Ricci flow, Kähler geometry · 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) 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.