A note on eigenvalue bounds for non-compact manifolds

2017/06/08 by Keller, Matthias, Liu, Shiping, Peyerimhoff, Norbert
#35P20 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1706.02437

Abstract

In this article we prove upper bounds for the Laplace eigenvalues λ_k 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.

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