As on arXiv
Abstract
This paper proves several topological results for smooth gradient Ricci shrinkers. We establish upper bounds for the Betti numbers, a vanishing theorem for cohomology, and a dichotomy for the number of ends. We also prove a full Hodge theorem for a large class of shrinkers. The methods are based on weighted cohomology and extend to self-shrinkers of the mean curvature flow.
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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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Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers
Topics Ricci solitons · Same author Fei He · Shared terms “Betti number”, “Ricci shrinkers”, “gradient Ricci”
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Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers
Topics Ricci solitons · Same author Fei He · Shared terms “Ricci shrinkers”, “upper bound”
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A stable self-shrinking Möbius bundle in
Topics Self-shrinkers & translators, Ricci solitons
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Closed embedded self-shrinkers of mean curvature flow
Topics Self-shrinkers & translators
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Gap theorems for complete self-shrinkers of -mean curvature flows
Topics Self-shrinkers & translators
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The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers
Topics Ricci solitons · Same author Fei He · Shared terms “Ricci shrinkers”, “upper bound”
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Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow
Topics Self-shrinkers & translators
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Convexity of 2-convex translating and expanding solitons to the mean curvature flow in
Topics Self-shrinkers & translators, Ricci solitons
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
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.