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On the spectrum of a finite-volume negatively-curved manifold
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 123, Number 2, April 2001
- pp. 185-205
- 10.1353/ajm.2001.0012
- Article
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We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to -1 and with an infinite number of gaps in the spectrum of the function Laplacian.