## Journal of Differential Geometry

### On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds

#### 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.

#### Article information

Source
J. Differential Geom., Volume 87, Number 3 (2011), 469-492.

Dates
First available in Project Euclid: 10 August 2011

https://projecteuclid.org/euclid.jdg/1312998232

Digital Object Identifier
doi:10.4310/jdg/1312998232

Mathematical Reviews number (MathSciNet)
MR2819545

Zentralblatt MATH identifier
1269.58011

#### Citation

Cianchi, Andrea; Maz’ya, Vladimir. On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds. J. Differential Geom. 87 (2011), no. 3, 469--492. doi:10.4310/jdg/1312998232. https://projecteuclid.org/euclid.jdg/1312998232