Small eigenvalues of closed Riemann surfaces for large genus

TL;DR

This paper investigates the asymptotic behavior of small eigenvalues of Riemann surfaces as the genus increases, showing they scale like 1/g^2 in the thick part of moduli space.

Contribution

It establishes uniform bounds for the small eigenvalues of high-genus Riemann surfaces and constructs specific surfaces with controlled geometric properties.

Findings

Small eigenvalues are asymptotically comparable to 1/g^2.

Constructed surfaces have boundary curves of fixed length in the thick part.

Number of separating systole curves scales linearly with 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 $g$.