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 \(\gamma \) and the area. Its proof relies on the existence of holomorhic maps to \(\mathbb {CP}^1\) of low degree. Very recently, Ros was able to use certain holomorphic maps to \(\mathbb {CP}^2\) in order to give a quantitative improvement of the Yang–Yau inequality for \(\gamma =3\). In the present paper, we generalize Ros’ argument to make use of holomorphic maps to \(\mathbb {CP}^n\) for any \(n>0\). As an application, we obtain a quantitative improvement of the Yang–Yau inequality for all genera except for \(\gamma = 4,6,8,10,14\).
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Acknowledgements
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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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
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DOI: https://doi.org/10.1007/s00209-022-03009-4