The Laplace operator on hyperbolic three manifolds with cusps of non-maximal rank

Abstract

where F is a geometrically finite discrete group of orientation preserving isometries, which may contain parabolic elements of non-maximal rank. We prove the existence of a continuum eigenfunction expansion, and show that in the spectral parameter the eigenfun- ctions have meromorphic continuations to 112\{ 1). These eigenfunctions are Eisen- stein series for F. The continuous spectrum of Laplace operators on hyperbolic manifolds was first analysed by number theorists, as a means of studying Eisenstein series. Maass and Roelke I-M, R] considered the case of finite volume manifolds in two dimen- sions. Increasingly more general hyperbolic manifolds were studied by Faddeev IF], Patterson [Pal, Lax and Phillips [LPI-LP4], Mandouvalos [M1-M3] and others. Recently Agmon l-A], Mazzeo and Melrose [MM], and Perry [P], proved the meromorphic continuation of the Eisenstein series for geometrically finite groups with fundamental domains of infinite volume provided the groups have no parabolic elements, or all the parabolic elements have maximal rank. In this case the boundary at infinity, which, roughly speaking, labels the directions at infinity in which scattering can occur, is a smooth compact manifold, or, in the presence of parabolic elements of maximal rank, the union of a smooth compact manifold with a finite number of points. In fact, the work of Mazzeo and Melrose and recent work of Agmon applies to more general manifolds which have a similar structure near infinity. However, none of the proofs of meromorphic continuation of eigenfun- ctions apply when the group has parabolic elements of non-maximal rank. We will study the Laplace operator on a three dimensional hyperbolic manifold when parabolic elements of non-maximal rank are present. In this case the * Research supported by the National Science and Engineering Research Council of Canada ** Research supported in part by N.S.F. grant DMS-8911242 *** Research supported in part by N.S.F. grant DMS-8802668