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Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\epsilon \)

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Abstract

Let \(\mathcal {M}_g\) be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Petersson metric. In this paper, we show that for any \(\epsilon >0\), as genus g goes to infinity, a generic surface \(X\in \mathcal {M}_g\) satisfies that the first eigenvalue \(\lambda _1(X)>\frac{3}{16}-\epsilon \). As an application, we also show that a generic surface \(X\in \mathcal {M}_g\) satisfies that the diameter \({{\,\mathrm{diam}\,}}(X)<(4+\epsilon )\ln (g)\) for large genus.

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Acknowledgements

The authors would like to thank Yang Shen for helpful discussions on the key counting result Theorem 4. They also would like to thank Prof. S. T. Yau for his interests on this work. Both authors are supported by the NSFC Grant No. 12171263, and the first named author is also partially supported by a grant from Tsinghua University. We are also grateful to the referees for helpful comments and suggestions which improve this article.

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  1. Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Yunhui Wu & Yuhao Xue

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  1. Yunhui Wu
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Correspondence to Yunhui Wu.

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Wu, Y., Xue, Y. Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\epsilon \). Geom. Funct. Anal. 32, 340–410 (2022). https://doi.org/10.1007/s00039-022-00595-7

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