Open Access
March 2011 On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds
Andrea Cianchi, Vladimir Maz’ya
J. Differential Geom. 87(3): 469-492 (March 2011). DOI: 10.4310/jdg/1312998232

Abstract

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold $M$ are established in terms of the isocapacitary function of $M$. The relevant capacity takes a different form according to whether $M$ has finite or infinite volume. Conditions involving the more standard isoperimetric function of $M$ can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.

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Andrea Cianchi. Vladimir Maz’ya. "On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds." J. Differential Geom. 87 (3) 469 - 492, March 2011. https://doi.org/10.4310/jdg/1312998232

Information

Published: March 2011
First available in Project Euclid: 10 August 2011

zbMATH: 1269.58011
MathSciNet: MR2819545
Digital Object Identifier: 10.4310/jdg/1312998232

Rights: Copyright © 2011 Lehigh University

Vol.87 • No. 3 • March 2011
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