As on arXiv
Abstract
We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining with recent work of Lott and Deruelle-Schulze-Simon gives a proof of Hamilton's pinching conjecture without additional hypotheses.
Comments: Accepted to The Journal of Differential Geometry
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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 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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PIC1 pinched manifolds are flat or compact
Topics Curvature pinching, 3-manifolds, Boundaries & non-compact · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “pinched Ricci”, “pinching conjecture”, “Hamilton pinching”
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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 “curvature decay”, “invariant curvature”, “combining”
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On locally conformally flat manifolds with positive pinched Ricci curvature
Topics Curvature pinching · Shared terms “pinched Ricci”, “pinching conjecture”, “Hamilton pinching”
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Uniqueness of Ricci flow with scaling invariant estimates
Topics 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “negatively”, “invariant curvature”, “complete Ricci”
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Time zero regularity of Ricci flow
Topics Ricci flow · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “positive time”
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A mean curvature flow approach to Hamilton's pinching theorem
Topics Curvature pinching · Shared terms “Hamilton pinching”
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Inverse mean curvature flow and Ricci-pinched three-manifolds
Topics 3-manifolds, Boundaries & non-compact · Shared terms “Deruelle”, “Schulze”, “Lott”
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.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.10Hamilton–Ivey Pinching
Why high curvature in dimension 3 is almost positive.