As on arXiv
Abstract
In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity – a noncollapsed -limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed -limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature (), either it is compact, or every tangent flow at infinity is a Ricci flat cone.
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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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A gap theorem for metric solitons and its applications
Topics Ancient solutions, Singularities · Same authors Ganqi Wang, Yongjia Zhang · Shared terms “limit metric”, “volume ratio”, “asymptotic volume”
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On noncollapsed -limit metric solitons
Topics (Non)collapsing, Singularities · Same author Yongjia Zhang · Shared terms “limit metric”, “noncollapsed”
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Topics Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “tangent flow”, “dichotomy theorem”
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Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “tangent flow”
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Dimension Reduction for Positively Curved Steady Solitons
Topics (Non)collapsing, Singularities · Same author Yongjia Zhang · Shared terms “noncollapsed”, “positively”, “tangent flow”
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Rigidity of positively curved Steady gradient Ricci solitons on orbifolds
Topics (Non)collapsing, Ancient solutions, Singularities · Same author Yuxing Deng · Shared terms “positively”, “noncollapsed”
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An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons
Topics (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “noncollapsed”
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On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature
Topics (Non)collapsing · Same author Yuxing Deng · 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.
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.
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.