As on arXiv
Abstract
We prove a local gap theorem for Ricci shrinkers, which states that if the local -functional at scale on a large ball centered at the minimum point of the potential function is close enough to , then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global -functional to be close enough to . Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].
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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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Rigidity and gap theorems for Ricci shrinkers
Topics Ricci solitons · Same authors Pak-Yeung Chan, Yongjia Zhang · Shared terms “gap theorem”, “local gap”, “Ricci shrinkers”
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On noncollapsed -limit metric solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “local gap”, “gap theorem”, “Ricci shrinkers”
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A gap theorem for metric solitons and its applications
Same author Yongjia Zhang · Shared terms “gap theorem”, “close enough”
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A local Sobolev inequality on Ricci flow and its applications
Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “local geometry”
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Hamilton-Ivey estimates for gradient Ricci solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “potential function”
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Volume growth estimates of gradient Ricci solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang
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Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang
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Dimension Reduction for Positively Curved Steady Solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang
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
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 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.