Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consisting of those vector fields V on X satisfying V x = O(x 2) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the ‘small calculus ’ of pseudodifferential operators Ψ ∗ Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an ‘exact fibred cusp ’ metric is analyzed as is the wavefront set associated to the calculus.

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We compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.

In the first part of my talk I shall describe some of the properties one should expect of a calculus of pseudodifferential operators which corresponds to the microlocalization of a Lie algebra of vector fields. This is not intended to be a formal axiomatic program but it leads one to consider conditions on the Lie algebra for such microlocalization to be possible. The symbolic structure of the calculus also shows how it can be applied in the solution of analytic questions related to the Lie algebra, especially to the inversion of elliptic elements of the enveloping algebra. In the second part I shall consider several such analytic question which arise in various differential–geometric settings and are, or appear to be, amenable to the application of these pseudodifferential techniques. For those examples which have already been analyzed the Lie algebra is identified and then a specific question is discussed using the calculus of pseudodifferential operators which arises from it. Finally in the third part of the talk I shall briefly outline a general strategy for the construction of ‘the ’ calculus of pseudodifferential operators which microlocalizes a given Lie algebra satisfying conditions which make it a boundary–fibration

Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.

Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.

this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex co-compact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110

Several interesting examples of non-compact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie algebras of vector fields – baptised “manifolds with a Lie structure at infinity ” there – was studied from an axiomatic point of view. In this paper, we define and study the algebra Ψ ∞ 1,0,V (M0), which is an algebra of pseudodifferential operators canonically associated to a manifold M0 with the Lie structure at infinity V ⊂ Γ(M; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M0). We also consider the algebra Diff ∗ V (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞ 1,0,V (M0) is a “microlocalization” of Diff ∗ V (M0). We also define and study semi-classical and “suspended ” versions of the algebra Ψ ∞ 1,0,V (M0). Thus, our constructions solves a conjecture of Melrose [28].

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown

this paper is to prove the conjecture of Patterson for convex cocompact #