A note on eigenvalue bounds for non-compact manifolds

Matthias Keller; Shiping Liu; Norbert Peyerimhoff

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arXiv · 1706.02437

A note on eigenvalue bounds for non-compact manifolds

Matthias Keller·Shiping Liu·Norbert Peyerimhoff

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Abstract

In this article we prove upper bounds for the Laplace eigenvalues $\lambda_k$ below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of $k^2$ and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to $-\infty$, 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.

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Matthias Keller, Shiping Liu, Norbert Peyerimhoff. 2017-06-08. A note on eigenvalue bounds for non-compact manifolds. https://arxiv.org/abs/1706.02437

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