Paper · May 16, 2023

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

Wangjian Jian, Yalong Shi

math.DGarXiv:2305.09421
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Abstract

Assuming Perelman's estimates, we give a new proof of uniform estimate along normalized Kähler-Ricci flow on Fano manifolds with Kähler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle. This proof does not use pluripotential theory.

Comments: comments are welcome

Ricci flowKähler–Ricci flowEinstein metricsKähler geometryKähler–Einstein & K-stability

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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 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 K-stability of Gorenstein Fano group compactifications with rank two (Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo, 2022)8Lee 2022 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)5Zhao 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)1Jian 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)7Cho 2024 Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow (Alexander Bednarek, 2025)2Bednarek 2025 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)6Fong 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)3Jian 2026 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)4Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Wangjian Jian · Shared terms “Fano manifold”, “Perelman”

  2. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek PDF

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

  3. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Wangjian Jian · Shared terms “Fano”

  4. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “Fano manifold”

  5. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Chen Cheng”, “Ampère equation”, “Monge Ampère”

  6. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  7. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi PDF

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

  8. K-stability of Gorenstein Fano group compactifications with rank two

    Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo PDF

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

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. 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. 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.