A note on eigenvalue bounds for non‐compact manifolds

Matthias Keller, Shiping Liu and Norbert Peyerimhoff

Mathematische Nachrichten, 2021, vol. 294, issue 6, 1134-1139

Abstract: In this article we prove upper bounds for the Laplace eigenvalues λk below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k2 and specific geometric data of the manifold. This applies also to the particular case of non‐compact manifolds whose sectional curvature tends to −∞, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

Date: 2021
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https://doi.org/10.1002/mana.201900209

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