As on arXiv
Abstract
In this work, we construct distance like functions with integral hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors -regular and small curvature concentration are topologically Euclidean.
Comments: 31 pages
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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 note on a diffeomorphism criterion via long-time Ricci flow
Topics Scalar curvature · Same author Shaochuang Huang · Shared terms “small curvature”, “curvature concentration”
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Sharpening a gap theorem: nonnegative Ricci and small curvature concentration
Topics Ricci flow · Shared terms “curvature concentration”, “small curvature”
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A note on Ricci flow from small curvature concentration and a Morrey-type condition
Topics Ricci flow · Shared terms “small curvature”, “curvature concentration”, “bounded curvature”
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Gap Theorem on Riemannian manifolds using Ricci flow
Topics Ricci flow · Same authors Pak-Yeung Chan, Man-Chun Lee
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Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass
Topics Scalar curvature · Same author Man-Chun Lee · Shared terms “lower bound”, “Euclidean”
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Higher-dimensional flying wing Steady Ricci Solitons
Topics Ricci flow · Same authors Pak-Yeung Chan, Man-Chun Lee
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Expanding Ricci solitons coming out of weakly PIC1 metric cones
Topics Ricci flow · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “Euclidean”
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Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Scalar curvature · Same author Pak-Yeung Chan · Shared terms “lower bound”
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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
- 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.