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Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds
Abstract We show that for a closed hyperbolic 3‐manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1‐forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3‐manifolds with injectivity radius bounded below and volume going to infinity for which the 1‐form Laplacian has spectral gap vanishing exponentially fast in the volume.
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- Award ID(s):
- 1811156
- PAR ID:
- 10419925
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 16
- Issue:
- 2
- ISSN:
- 1753-8416
- Page Range / eLocation ID:
- p. 588-633
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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