As on arXiv
Abstract
We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of , , (), , all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of lies in the interior of our curvature cone.
- Submitted
- Nov 15, 2016
- Last revised
- Nov 13, 2017 (v3)
- Subjects
- math.DG
Comments: Ann. of Math. 187, 263-299 (2018)
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 2016, plus the canon.
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Ricci flow with surgery on manifolds with positive isotropic curvature
Topics Surgery, Curvature pinching, Singularities · Same author S. Brendle · Shared terms “surgery procedure”, “pinching estimate”, “dimensional version”
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Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow
Topics Curvature pinching · Shared terms “curvature cone”, “neighborhood theorem”, “canonical neighborhood”
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Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions
Topics Rotational symmetry & warped products · Same author S. Brendle · Shared terms “higher dimensions”
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Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
Topics Rotational symmetry & warped products · Same author S. Brendle · Shared terms “higher dimensions”, “rotationally symmetric”
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A canonical neighborhood theorem for the mean curvature flow in higher codimension
Topics Curvature pinching · Shared terms “neighborhood theorem”, “canonical neighborhood”
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High Codimension Mean Curvature Flow with Surgery
Topics Surgery, Curvature pinching, Singularities · Shared terms “connected sum”, “preserved”
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Uniqueness of compact ancient solutions to the higher dimensional Ricci flow
Topics Singularities · Same author S. Brendle · Shared terms “higher dimensional”, “Perelman”
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Canonical surgeries in rotationally invariant Ricci flow
Topics Surgery, Rotational symmetry & warped products, Singularities
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.10Surgery on a δ-Neck
The gluing construction in detail: the profile function, preservation of the Hamilton–Ivey pinching, closeness to the standard solution, and the definition of Ricci flow with surgery as a space-time (Morgan–Tian chapters 13–15).
- 12B.5The Canonical Neighbourhood Theorem
Every high-curvature point looks like a neck or a cap.
- 12B.7The Singular Time: Horns and Strong Necks
The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.