Small eigenvalues of closed Riemann surfaces for large genus

• Published in: Transactions of the American Mathematical Society 2022-05, Vol.375 (5), p.3641
• Main Authors: Wu, Yunhui, Xue, Yuhao
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/8608

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.