A note on eigenvalue bounds for non‐compact manifolds. Issue 6 (10th April 2021)

• Record Type: Journal Article
• Authors: Keller, Matthias; Liu, Shiping; Peyerimhoff, Norbert
• Is Part Of: Mathematische Nachrichten. Volume 294:Issue 6(2021)
• Page Start: 1134
• Page End: 1139
• Publication Date: 2021-04-10
• Subjects: Cheeger inequality -- eigenvalues -- Laplacian -- negative curvature -- Riemannian manifold; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900209
• Languages: English
• ISSNs: 0025-584X

Abstract

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.