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 $g$.

Transactions of the American Mathematical Society, to appear

References in corpus (2)

• Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus
• Growth of the Weil-Petersson inradius of moduli space

Cited by in corpus (4)

• Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$
• Large genus asymptotics for lengths of separating closed geodesics on random surfaces
• Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus
• Degenerating hyperbolic surfaces and spectral gaps for large genus