A note on eigenvalue bounds for non-compact manifolds

2017/06/08 by Matthias Keller, Shiping Liu, Keller, Matthias +3
Mathematics · #35P20 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:35P20 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1706.02437

8 pages. To appear in Math. Nachr

arxiv created 2020/07/16 · arxiv updated 2020/07/17

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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