On the existence of Maass cusp forms on hyperbolic surfaces with cone points

Abstract

The perturbation theory of the Laplace spectrum of hyperbolic surfaces with conical singularities belonging to a fixed conformal class is developed. As an application, it is shown that the generic such surface with cusps has no Maass cusp forms ( L 2 {L^2} eigenfunctions) under specific eigenvalue multiplicity assumptions. It is also shown that eigenvalues depend monotonically on the cone angles. From this, one obtains Neumann eigenvalue monotonicity for geodesic triangles in H 2 {{\mathbf {H}}^2} and a lower bound of 1 2 π 2 \frac {1}{2}{\pi ^2} for the eigenvalues of ‘odd’ Maass cusp forms associated to Hecke triangle groups.