Small eigenvalues of closed Riemann surfaces for large genus

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 k , as the genus g g goes to infinity, the minimum of k k -th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus g g is uniformly comparable to 1 g 2 \frac {1}{g^2} in g g . And the minimum of a g ag -th eigenvalues of hyperbolic surfaces in any thick part of moduli space is bounded above by a uniform constant only depending on ε \varepsilon and a a . In the proof of the upper bound, for any constant ε > 0 \varepsilon >0 , we will construct a closed hyperbolic surface of genus g g in any ε \varepsilon -thick part of moduli space such that it admits a pants decomposition whose curves all have length equal to ε \varepsilon , and the number of separating systole curves in this surface is uniformly comparable to g g .