Paper · Jan 17, 2020

On the Kahler-Ricci flow on Fano manifolds

Bin Guo, Duong H. Phong, Jacob Sturm

math.DGarXiv:2001.06329
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As on arXiv

Abstract

A short proof of the convergence of the Kahler-Ricci flow on Fano manifolds admitting a Kahler-Einstein metric or a Kahler-Ricci soliton is given, using a variety of recent techniques

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Submitted
Jan 17, 2020
Subjects
math.DG
Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–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 Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)6Hultgren 2017 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)1Guo 2018 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)5Liu 2019 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)2Wang 2020 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)7Wang 2023 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)4Jian 2023 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)3Nitta 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)8Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compactness of Kahler-Ricci solitons on Fano manifolds

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Same authors Bin Guo, Duong H. Phong, Jacob Sturm · Shared terms “Fano manifolds”

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

    Feng Wang and Xiaohua Zhu

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

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

    Yasufumi Nitta

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

  4. 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”

  5. 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 · Shared terms “Fano manifolds”

  6. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

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

  7. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

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

  8. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · 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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

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.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.