Paper · Oct 27, 2022

A Note On Kähler-Ricci Flow on Fano Threefolds

Minghao Miao, Gang Tian

math.DGarXiv:2210.15263Peking Math. J., 8(1):191-199, 2025
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Abstract

In this note, we show that the solution of Kähler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits Kähler-Ricci soliton and the Gromov-Hausdorff limit of the Kähler-Ricci flow must be a singular -Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which Kähler-Ricci flow develops type II singularity.

Comments: 6 pages, comments are welcome!

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularities3-manifoldsKähler geometryLimit spaces & RCD

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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 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 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)8Tian 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)1Tian 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)4Wang 2023 Kähler-Ricci solitons on Fano threefolds with non-trivial moduli (Minghao Miao and Linsheng Wang, 2023)3Miao 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)5Hallgren 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)6Jian 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)7Jian 2023 Optimal Degenerations of K-unstable Fano threefolds (Minghao Miao and Linsheng Wang, 2024)2Miao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Limit spaces & RCD · Same author Gang Tian · Shared terms “Fano”, “Hausdorff limit”, “Gromov Hausdorff”

  2. Optimal Degenerations of K-unstable Fano threefolds

    Minghao Miao and Linsheng Wang PDF

    Topics Kähler–Ricci solitons, 3-manifolds, Kähler geometry · Same author Minghao Miao · Shared terms “Fano threefold”, “Mori Mukai”, “Mukai list”

  3. Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

    Minghao Miao and Linsheng Wang PDF

    Topics Kähler–Ricci solitons, 3-manifolds, Kähler geometry · Same author Minghao Miao · Shared terms “Fano threefold”, “Fano varieties”

  4. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Limit spaces & RCD · Shared terms “Fano”, “Hausdorff limit”, “Gromov Hausdorff”

  5. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “Gromov Hausdorff”

  6. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “develop Type”, “Fano”, “Gromov Hausdorff”

  7. 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, Singularities · Same author Gang Tian · Shared terms “Fano”

  8. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Gang Tian · 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.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.