Bessa, G. Pacelli and Gimeno, Vincent and Polymerakis, Panagiotis (2021) The spectrum of the Laplacian and volume growth of proper minimal submanifolds. MPIM Preprint Series 2021 (15).
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Abstract
We give upper bounds for the bottom of the essential spectrum of properly immersed minimal submanifolds of $\mathbb{R}^n$ in terms of their volume growth. This result can be viewed as an extrinsic version of Brook's essential spectrum estimate [6, Thm.1] and it gives a fairly general answer to a question of S. T. Yau [26, p.240] about what upper bounds for the first eigenvalue (bottom of the spectrum) of immersed minimal surfaces of $\mathbb{R}^3$.
| Item Type: | MPIM Preprint |
|---|---|
| Subjects: | 3 Geometry and topology > 58-XX Global analysis, analysis on manifolds |
| Divisions: | Research > Preprints |
| Depositing User: | Andrea Kohlhuber |
| Date Deposited: | 18 May 2021 10:03 |
| Last Modified: | 18 May 2021 11:04 |
| URI: | https://archive.mpim-bonn.mpg.de/id/eprint/4545 |
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