A note on eigenvalue bounds for non‐compact manifolds. Issue 6 (10th April 2021) - Keller, Matthias; Liu, Shiping; Peyerimhoff, Norbert
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 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 − ∞, 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.