Karpukhin, Mikhail and Vinokurov, Denis (2022) The first eigenvalue of the Laplacian on orientable surfaces. Mathematische Zeitschrift, 301 (3). pp. 2733-2746. ISSN 0025-5874. doi:10.1007/s00209-022-03009-4. https://resolver.caltech.edu/CaltechAUTHORS:20220307-189685000
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Abstract
The famous Yang-Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus γ and the area. Its proof relies on the existence of holomorhic maps to CP¹ of low degree. Very recently, A.~Ros was able to use certain holomorphic maps to CP² in order to give a quantitative improvement of the Yang-Yau inequality for γ = 3. In the present paper, we generalize Ros' argument to make use of holomorphic maps to CP^n for any n > 0. As an application, we obtain a quantitative improvement of the Yang-Yau inequality for all genera γ > 3 except for γ = 4,6,8,10,14.
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Alternate Title: | An improved Yang-Yau inequality for the first Laplace eigenvalue | ||||||||||||
Additional Information: | © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022. Received: 7 July 2021 / Accepted: 26 January 2022. The authors would like to thank I. Polterovich and A. Penskoi for fruitful discussions. The first author is partially supported by NSF grant DMS-1363432. | ||||||||||||
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Issue or Number: | 3 | ||||||||||||
DOI: | 10.1007/s00209-022-03009-4 | ||||||||||||
Record Number: | CaltechAUTHORS:20220307-189685000 | ||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20220307-189685000 | ||||||||||||
Official Citation: | Karpukhin, M., Vinokurov, D. The first eigenvalue of the Laplacian on orientable surfaces. Math. Z. 301, 2733–2746 (2022). https://doi.org/10.1007/s00209-022-03009-4 | ||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
ID Code: | 113779 | ||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||
Deposited By: | George Porter | ||||||||||||
Deposited On: | 08 Mar 2022 21:17 | ||||||||||||
Last Modified: | 12 Jul 2022 18:16 |
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