Paper · Sep 22, 2020

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

Feng Wang, Xiaohua Zhu

math.DGarXiv:2009.10354
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Abstract

In this paper, we study the uniformly strong convergence of Kähler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in sense of diffeomorphism orbits. The result generalizes Tian-Zhu's theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun's result of for the uniqueness of of Kähler-Einstein metric orbits.

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Submitted
Sep 22, 2020
Subjects
math.DG
Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsEinstein 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 Kähler-Ricci flow and optimal degenerations (Ruadhaí Dervan and Gábor Székelyhidi, 2016)7Dervan 2016 Singular limits of Kähler-Ricci flow on Fano G-manifolds (Yan Li, Gang Tian and Xiaohua Zhu, 2018)6Li 2018 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)3Guo 2020 Tian's partial C⁰-estimate implies Hamilton-Tian's conjecture (Feng Wang and Xiaohua Zhu, 2020)2Wang 2020 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)4Tian 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)8Tian 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)1Wang 2023 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)5Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same authors Feng Wang, Xiaohua Zhu · Shared terms “Fano manifolds”

  2. Tian's partial -estimate implies Hamilton-Tian's conjecture

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same authors Feng Wang, Xiaohua Zhu · Shared terms “Fano manifolds”, “Tian”

  3. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

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

  4. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Xiaohua Zhu · Shared terms “complex structure”, “Fano manifolds”, “Tian”

  5. 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 · Same author Xiaohua Zhu · Shared terms “Fano manifolds”

  6. Singular limits of Kähler-Ricci flow on Fano -manifolds

    Yan Li, Gang Tian and Xiaohua Zhu

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

  7. The Kähler-Ricci flow and optimal degenerations

    Ruadhaí Dervan and Gábor Székelyhidi

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

  8. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · 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) 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.