Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus

Abstract

Abstract:In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than ${\cal L}_1(X_g)\over g^2$ up to a uniform positive constant multiplication. Where ${\cal L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g\to\infty$.