As on arXiv
Abstract
Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.
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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, 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 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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Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds
Topics 3-manifolds · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “pinched manifold”, “Hamilton”
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On the Hamilton-Lott conjecture in higher dimensions
Topics Sphere theorems, Boundaries & non-compact · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “PIC”, “dimensional complete”, “Hamilton”
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Ricci flow and PIC1
Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”
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Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass
Topics Boundaries & non-compact · Same authors Man-Chun Lee, Miles Simon
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The Harnack inequality without convexity for curve shortening flow
Topics Harnack inequalities · Same author Peter M. Topping · Shared terms “curvature estimates”, “alternative”, “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 “gap theorem”, “curvature estimates”
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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.
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.
- 6D.5Diffusions on Manifolds and Curvature
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.