Platonic surfaces
The main purpose of this paper is to show that it is possible to compare the hyperbolic metrics on two Riemann surfaces one of which is compact and the other is a punctured version of the same Riemann surface. Surprisingly this is possible outside a neighbourhood of the cusps, and it can be proved using the Ahlfors-Schwarz ``Lemma''. The author uses this to prove that if \(G_k\) uniformizes the compactification of \(\Gamma(k)\setminus{\mathbf H}\), \(k>6\), then the first non-trivial eigenvalue \(\lambda_1\) of the Laplace operator on \(G_k \setminus{\mathbf H}\) satisfies \(\lambda_1\geq 5/36\). This makes use of Selberg's 3/16 estimate in the case of the \(\Gamma(k)\). The author also gives a comparison of the Cheeger constants in the general situation.
scientific article; zbMATH DE number 1278817