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arXiv · 2102.05581

Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-ε$

Yunhui Wu·Yuhao Xue
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Abstract

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $ε>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $λ_1(X)>\frac{3}{16}-ε$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+ε)\ln(g)$ for large genus.

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Yunhui Wu, Yuhao Xue. 2022-03-29. Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-ε$. https://doi.org/10.1007/s00039-022-00595-7

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