Systole and small eigenvalues of hyperbolic surfaces

Abstract

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$\chi$, and let $\lambda_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $\lambda_{-\chi}>\frac14$. In this article, we prove that if thesystole of $S$ is greater than 3,46, then $\lambda_{-\chi-1}>\frac14$.This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.