On the discreteness of the spectrum of the laplacian on noncompact riemannian manifolds

Andrea Cianchi Universit`a di Firenze Vladimir Maz’ya Link¨oping University

Differential Geometry mathscidoc:1609.10213

Journal of Differential Geometry, 87, (3), 469-491, 2011
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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@inproceedings{andrea2011on,
  title={ON THE DISCRETENESS OF THE SPECTRUM OF THE LAPLACIAN ON NONCOMPACT RIEMANNIAN MANIFOLDS},
  author={Andrea Cianchi, and Vladimir Maz’ya},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160912071511843730872},
  booktitle={Journal of Differential Geometry},
  volume={87},
  number={3},
  pages={469-491},
  year={2011},
}
Andrea Cianchi, and Vladimir Maz’ya. ON THE DISCRETENESS OF THE SPECTRUM OF THE LAPLACIAN ON NONCOMPACT RIEMANNIAN MANIFOLDS. 2011. Vol. 87. In Journal of Differential Geometry. pp.469-491. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160912071511843730872.
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