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.
Citation
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
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