As on arXiv
Abstract
In this paper, we prove a gap theorem for -limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an -regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).
Comments: 27 pages
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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 Ricci flows with nonnegative Ricci curvature
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 Scalar curvature, Singularities · Same author Yongjia Zhang · Shared terms “limit metric”, “gap theorem”
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Rigidity and gap theorems for Ricci shrinkers
Topics Singularities · Same author Yongjia Zhang · Shared terms “gap theorem”, “Wang”
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On Ricci flows with closed and smooth tangent flows
Topics Ancient solutions, Singularities · Same author Yongjia Zhang
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A local gap theorem for Ricci shrinkers
Same author Yongjia Zhang · Shared terms “gap theorem”, “close enough”
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A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II
Topics Ancient solutions, Scalar curvature, Singularities · Shared terms “curvature bounds”
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Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Scalar curvature, Singularities · Same author Yongjia Zhang
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An -regularity theorem for Perelman's reduced volume
Topics Ricci flow · Same author Yongjia Zhang · Shared terms “regularity theorem”, “curvature radius”
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.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.