As on arXiv
Abstract
In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence theory without non-collapsing assumption. We use it to show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay are necessarily flat. The second main result concerns three-manifolds with non-negative Ricci curvature of quadratic decay. By combining our newly established curvature estimate and method in Kähler geometry, we show that if the curvature decays slightly faster even in average sense, the manifold must be flat. This strengthens a result of Reiris. In the compact case, we use the Ricci flow regularization to generalize the celebrated Gromov-Ruh Theorem in this direction.
Comments: 35 pages, minor changes
Its neighbourhood
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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Higher-dimensional flying wing Steady Ricci Solitons
Topics (Non)collapsing, Boundaries & non-compact · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “negative curvature”, “scaling invariant”
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Uniqueness of Ricci flow with scaling invariant estimates
Topics (Non)collapsing, 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “scaling invariant”, “invariant curvature”
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Manifolds with PIC1 pinched curvature
Topics 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “negative complex”, “complex sectional”, “sectional curvature”
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Ricci Flow under Kato-type curvature lower bound
Topics (Non)collapsing, 3-manifolds · Same author Man-Chun Lee · Shared terms “existence theory”
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Three-manifolds with non-negatively pinched Ricci curvature
Topics 3-manifolds, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “curvature decay”, “invariant curvature”, “combining”
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Expanding Ricci solitons coming out of weakly PIC1 metric cones
Topics Kähler geometry · Same authors Pak-Yeung Chan, Man-Chun Lee
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Gap Theorem on locally conformally flat manifold
Same author Man-Chun Lee · Shared terms “gap theorem”, “curvature decay”, “gap phenomenon”
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Manifolds with small curvature concentration
Topics Ricci flow · Same authors Pak-Yeung Chan, 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
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
- 12A.4κ-Noncollapsing
The theorem that excludes the cigar.
- 12A.8Local Noncollapsing from the Reduced Volume
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.