As on arXiv
Abstract
In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the norm for certain of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.
Comments: To appear in the Vaughan Jones Memorial Volume of the NZJM
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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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A gap theorem for metric solitons and its applications
Topics Scalar curvature · Shared terms “gap theorem”
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A topological gap theorem for the -systole of positive scalar curvature 3-manifolds
Topics Scalar curvature · Shared terms “gap theorem”, “positive scalar”
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Limit theorems for the total scalar curvature
Topics Scalar curvature · Shared terms “Riemannian metrics”
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Weak scalar curvature lower bounds along Ricci flow
Topics Scalar curvature · Shared terms “Yamabe”
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On weak formulations of (super) Ricci flows
Topics Scalar curvature · Shared terms “positive scalar”
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Singular Ricci Flows on surfaces with boundary and positive scalar curvature
Topics Scalar curvature · Shared terms “positive scalar”
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On the geometry of Riemannian warped product maps
Topics Scalar curvature · Shared terms “curvature tensor”
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Manifolds with small curvature concentration
Topics Scalar 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
- 6C.4The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
- 9B.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.