As on arXiv
Abstract
We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.
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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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Manifolds with PIC1 pinched curvature
Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”
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PIC1 pinched manifolds are flat or compact
Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”
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Manifolds with 1/4-pinched Curvature are Space Forms
Topics Sphere theorems · Shared terms “Riemannian manifolds”
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Steady Gradient Ricci Solitons with Symmetry
Topics Sphere theorems · Shared terms “positive curvature”
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Expanding Ricci solitons coming out of weakly PIC1 metric cones
Topics Sphere theorems · Shared terms “PIC”
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Rigidity of area non-increasing maps
Topics Sphere theorems · Shared terms “positive curvature”
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
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.