Paper · Aug 16, 2020

Ancient solutions to the Kähler Ricci flow

Yu Li

math.DGarXiv:2008.06951v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We give a complete classification of all -noncollapsed, complete ancient solutions to the Kähler Ricci flow with nonnegative bisectional curvature.

View PDF View HTML arXiv page
Submitted
Aug 16, 2020
Last revised
Dec 3, 2024 (v2)
Subjects
math.DG
Journal
Geom. Topol. 28 (2024) 3257-3283
DOI
10.2140/gt.2024.28.3257

Comments: Revised version: some results have been improved, and their corresponding proofs have been simplified

Ricci flowKähler–Ricci flowAncient solutions(Non)collapsingKähler geometry

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 Ancient Solutions on Bundles with Non-abelian Structural Group (Peng Lu and Y. K. Wang, 2016)7Lu 2016 On existence and curvature estimates of Ricci flow (Man-Chun Lee and Luen-Fai Tam, 2017)4Lee 2017 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature (Xiaolong Li and Lei Ni, 2019)2Li 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)5Brendle 2020 Ancient solutions to the Ricci flow with isotropic curvature conditions (Jae Ho Cho and Yu Li, 2020)1Cho 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)6Brendle 2021 Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers (Yu Li and Junsheng Zhang, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient solutions to the Ricci flow with isotropic curvature conditions

    Jae Ho Cho and Yu Li

    Topics Kähler–Ricci flow, (Non)collapsing, Ancient solutions · Same author Yu Li · Shared terms “complete ancient”, “noncollapsed”

  2. Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature

    Xiaolong Li and Lei Ni

    Topics Kähler–Ricci flow, Ancient solutions, Kähler geometry · Shared terms “bisectional curvature”

  3. Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions · Shared terms “complete ancient”

  4. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “nonnegative bisectional”, “bisectional curvature”

  5. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

  6. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

  7. Ancient Solutions on Bundles with Non-abelian Structural Group

    Peng Lu and Y. K. Wang

    Topics (Non)collapsing, Ancient solutions, Kähler geometry · Shared terms “noncollapsed”

  8. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

    Yu Li and Junsheng Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Yu Li · Shared terms “noncollapsed”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.