As on arXiv
Abstract
Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.
Comments: To appear, Geometry and Topology
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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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PIC1 pinched manifolds are flat or compact
Topics Sphere theorems, 3-manifolds, Boundaries & non-compact · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “PIC pinched”, “pinched Ricci”, “dimensional complete”
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Three-manifolds with non-negatively pinched Ricci curvature
Topics 3-manifolds, Boundaries & non-compact · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “negatively pinched”, “pinched Ricci”, “Deruelle Schulze”
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Ricci flow and PIC1
Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”
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Expanding Ricci solitons coming out of weakly PIC1 metric cones
Topics Sphere theorems · Same author Man-Chun Lee · Shared terms “PIC”, “Deruelle Schulze”, “Schulze Simon”
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Manifolds with 1/4-pinched Curvature are Space Forms
Topics Sphere theorems · Shared terms “pinched curvature”, “sectional curvature”, “Hamilton”
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Gap Theorem on Riemannian manifolds using Ricci flow
Topics 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “negative complex”, “complex sectional”, “sectional curvature”
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On the Hamilton-Lott conjecture in higher dimensions
Topics Sphere theorems, Boundaries & non-compact · Shared terms “Lott”, “PIC”, “dimensional complete”
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Rigidity of area non-increasing maps
Topics Sphere theorems · Same author Man-Chun Lee
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.
- 10A.1A Zoo of Three-Manifolds
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
- 10A.2Finite Rotation Groups and Spherical Space Forms
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.