Paper · Jul 24, 2026
Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers
As on arXiv
Abstract
We study the spectrum of the -Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the -th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to -Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.
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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
- 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.