The canon · Paper · Sep 7, 2020
Structure theory of non-collapsed limits of Ricci flows
As on arXiv
Abstract
Bamler III. Structure theory of non-collapsed limits of Ricci flows in all dimensions.
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension . We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected -curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelman's results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications.
Comments: 155 pages; Comments and questions welcome; improved introduction and minor edits
Its neighbourhood
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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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.
Foundational paper in the 4-manifolds guide: Bamler: structure theory, which describes 4D singularity formation in general.
Foundational paper in the Limit spaces & RCD guide: Bamler: structure of non-collapsed limits of Ricci flows.
Foundational paper in the Singularities guide: Bamler: structure of non-collapsed limits and singularity formation in all dimensions.
In the guidebook
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
- 12A.4κ-Noncollapsing
The theorem that excludes the cigar.
- 12A.8Local Noncollapsing from the Reduced Volume
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.