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arXiv:2207.12919 [math.GT]AbstractReferencesReviewsResources

On second eigenvalues of closed hyperbolic surfaces for large genus

Yuxin He, Yunhui Wu

Published 2022-07-26Version 1

In this article the second eigenvalues of closed hyperbolic surfaces for large genus have been studied. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, the second eigenvalue $\lambda_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$ up to uniform positive constants multiplications; moreover these two bounds are optimal as $g\to \infty$. Where $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also study the quantity $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$.

Comments: 37 pages, 6 figures, comments are welcome
Categories: math.GT, math.DG, math.SP
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