Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus

In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than ${\cal L}_1(X_g)\over g^2$ up to a uniform positive constant multiplication. Where ${...

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Bibliographic Details
Published in:American journal of mathematics 2022-08, Vol.144 (4), p.1087-1114
Main Authors: Wu, Yunhui, Xue, Yuhao
Format: Article
Language:English
Subjects:
Eigenvalues
Lower bounds
Riemann surfaces
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Summary:In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than ${\cal L}_1(X_g)\over g^2$ up to a uniform positive constant multiplication. Where ${\cal L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g\to\infty$.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2022.0024