Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces

Abstract

LetM be a complete Riemannian surface with constant curvature −1, infinite volume, and a finitely generated fundamental group. Denote by λ(M) the lowest eigenvalue of the Laplacian onM, and let Φ _M be the associated eigenfunction. We estimate the size of λ(M) and the shape of Φ _M by a finite procedure which has an electrical circuit analogue. Using the Margulis lemma, we decomposeM into its thick and thin parts. On the compact thick components, we show that Φ _M varies from a constant value by no more thanO($\sqrt {\lambda (M)}$). The estimate for λ(M) is calculable in terms of the topology ofM and the lengths of short geodesics ofM. An analogous theorem of the compact case was treated in [SWY].