Bas du spectre de surfaces hyperboliques de volume infini

Samuel Tapie (1)

1 IF - Institut Fourier

This article presents some methods to control the bottom of the spectrum of the Laplacian $\lambda_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $\lambda_0$ of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built by gluing copies of a geometrically finite surface with boundary according to the plan of an infinite graph. We control the $\lambda_0$ of the so-obtained infinite surfaces by constants coming from spectral properties of the building brick and combinatorial datae of the graph. We then use these methods to control the $\lambda_0$ of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings.

Mots clés: spectrum; lowest eigenvalue; Laplacian; hyperbolic geometry; infinite type surfaces; convex core

Domaines: Géométrie différentielle [math.DG]

Samuel Tapie. Bas du spectre de surfaces hyperboliques de volume infini. 2008. ⟨hal-00305887⟩