Small eigenvalues of closed Riemann surfaces for large genus

Abstract: In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. In the proof of the upper bound, for any constant $\epsilon>0$, we will construct a closed Riemann surface of genus $g$ in any $\epsilon$-thick part of moduli space such that it admits a pants decomposition whose boundary curves all have length equal to $\epsilon$, and the number of separating systole curves in this surface is uniformly comparable to $g$.