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The first eigenvalue of the Laplacian on orientable surfaces

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00209-022-03009-4DOIArticle
https://rdcu.be/cIys6PublisherFree ReadCube access
https://arxiv.org/abs/2106.00627arXivDiscussion Paper
ORCID:
AuthorORCID
Karpukhin, Mikhail0000-0003-1935-731X
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.
Funders:
Funding AgencyGrant Number
NSFDMS-1363432
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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