Platonic surface

Abstract

If SO is a Riemann surface with a complete metric of finite area and constant curvature -1, let SC denote the conformal compactification of SO. We show that, under the assumption that the cusps of SO are large, there is a close relationship between the hyperbolic metrics on SO and SC. We use this relationship to show that ${\mathrm{liminf}}_{k\to \infty }{\lambda }_1(P_k)\ge 5/36$, where the Platonic surface Pk is the conformal compactification of the modular surface Sk.