## Small eigenvalues of closed Riemann surfaces for large genus

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- by Yunhui Wu and Yuhao Xue PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3641-3663 Request permission

## Abstract:

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the minimum of $k$-th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac {1}{g^2}$ in $g$. And the minimum of $ag$-th eigenvalues of hyperbolic surfaces in any thick part of moduli space is bounded above by a uniform constant only depending on $\varepsilon$ and $a$.

In the proof of the upper bound, for any constant $\varepsilon >0$, we will construct a closed hyperbolic surface of genus $g$ in any $\varepsilon$-thick part of moduli space such that it admits a pants decomposition whose curves all have length equal to $\varepsilon$, and the number of separating systole curves in this surface is uniformly comparable to $g$.

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## Additional Information

**Yunhui Wu**- Affiliation: Tsinghua University, Haidian District, Beijing 100084, People’s Republic of China
- Email: yunhui_wu@mail.tsinghua.edu.cn
**Yuhao Xue**- Affiliation: Tsinghua University, Haidian District, Beijing 100084, People’s Republic of China
- ORCID: 0000-0002-0621-1007
- Email: xueyh18@mails.tsinghua.edu.cn
- Received by editor(s): April 10, 2019
- Received by editor(s) in revised form: November 10, 2021
- Published electronically: February 24, 2022
- Additional Notes: Both authors were supported by the NSFC grant No. 12171263.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3641-3663 - MSC (2020): Primary 35P15, 30F60, 58J50
- DOI: https://doi.org/10.1090/tran/8608
- MathSciNet review: 4402671