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First eigenvalue of the Laplacian on compact surfaces for large genera

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Abstract

For any Riemannian metric \(ds^2\) on a compact surface of genus g, Yang and Yau proved that the normalized first eigenvalue of the Laplacian \(\lambda _1(ds^2)Area(ds^2)\) is bounded in terms of the genus. In particular, if \(\Lambda _1(g)\) is the supremum for each g, it follows that the asymptotic growth of the sequence \({\Lambda _1(g)}\) is no larger than the one of \(4\pi g\). Recently Ros, for \(g=3\), and Karpukhin and Vinokurov, for the general case, improve these bounds. In this paper we obtain a sharper result for \(\Lambda _1(g)\) and we show that

$$\begin{aligned} \limsup _{g\, \rightarrow \, \infty } \, \frac{1}{g}\Lambda _1(g) \le 4(3-\sqrt{5})\pi \approx 3.056\pi . \end{aligned}$$

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  1. Department of Geometry and Topology and Institute of Mathematics (IMAG), University of Granada, 18071, Granada, Spain

    Antonio Ros

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  1. Antonio Ros
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This work is supported in part by the IMAG-Maria de Maeztu Grant CEX2020-001105-M / AEI / 10.13039/501100011033, MICINN Grant PID2020-117868GB-I00 and Junta de Andalucıa Grant P18-FR4049.

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Ros, A. First eigenvalue of the Laplacian on compact surfaces for large genera. Math. Z. 305, 62 (2023). https://doi.org/10.1007/s00209-023-03382-8

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