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Abstract
The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension . More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension is modeled on an ancient -solution. Moreover, Perelman proved a structure theorem for ancient -solutions in dimension . In this survey, we discuss recent developments which have led to a complete classification of all the singularity models in dimension . Moreover, we give an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension (originally proved by the author in 2012).
Comments: To appear in KIAS Springer Series in Mathematics, vol 1
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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.
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Ancient solutions to the Ricci flow in dimension 3
Topics Ancient solutions, 3-manifolds, Singularities · Same author S. Brendle · Shared terms “time singularity”, “Perelman”, “finite time”
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Ricci flow on manifolds with positive isotropic curvature in all dimensions
Topics (Non)collapsing, Ancient solutions, Ricci solitons · Same author S. Brendle · Shared terms “noncollapsed”
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Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “noncollapsed steady”, “steady gradient”, “Perelman”
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Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set
Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “steady gradient”, “noncollapsed steady”, “gradient Ricci”
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On Ricci flows with closed and smooth tangent flows
Topics Ancient solutions, Singularities · Shared terms “singularity models”, “time singularity”, “finite time”
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Ancient solutions of Ricci flow with Type I curvature growth
Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “singularity formation”, “noncollapsed”
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O(2)-symmetry of 3D steady gradient Ricci solitons
Topics Ancient solutions, 3-manifolds, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”
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Structure theory of non-collapsed limits of Ricci flows
Topics (Non)collapsing, 3-manifolds, Singularities · Shared terms “singularity formation”, “Perelman”
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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12B.4Compactness and Universal Estimates for κ-Solutions
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.
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
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.