As on arXiv
Abstract
We introduce a classification conjecture for -solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of -symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.
Comments: 12 pages (v2: added details in Section 2)
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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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Free boundary flow through cylindrical singularities
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Topics Ancient solutions, 3-manifolds, Weak & singular flows · Same author Robert Haslhofer · Shared terms “Perelman”
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Topics 3-manifolds, Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “canonical neighborhood”
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Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “sheet oval”, “bubble sheet”, “parameter family”
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Topics Weak & singular flows, Singularities, Stability & uniqueness · Same author Robert Haslhofer · Shared terms “neighborhood theorem”, “convex neighborhood”, “cylindrical singularities”
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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
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 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.