A note on eigenvalue bounds for non-compact manifolds
- 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 -infinity, 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.
| Author details: | Matthias KellerORCiDGND, Shiping Liu, Norbert PeyerimhoffORCiDGND |
|---|---|
| DOI: | https://doi.org/10.1002/mana.201900209 |
| ISSN: | 0025-584X |
| ISSN: | 1522-2616 |
| Title of parent work (English): | Mathematische Nachrichten |
| Publisher: | Wiley-VCH |
| Place of publishing: | Weinheim |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2021/04/10 |
| Publication year: | 2021 |
| Release date: | 2024/05/29 |
| Tag: | Cheeger inequality; Laplacian; Riemannian manifold; eigenvalues; negative curvature |
| Volume: | 294 |
| Issue: | 6 |
| Number of pages: | 6 |
| First page: | 1134 |
| Last Page: | 1139 |
| Funding institution: | Deutsche ForschungsgemeinschaftGerman Research Foundation (DFG) |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
| Publishing method: | Open Access / Hybrid Open-Access |
| License (German): |  CC-BY-NC-ND - Namensnennung, nicht kommerziell, keine Bearbeitungen 4.0 International |
