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.

Abstract. We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption. Contents

Abstract. We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.

We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering

We apply our previous work on Green’s functions for the four–dimensional quaternionic Taub–NUT manifold to obtain a scalar two–point function on the homogeneously squashed three–sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal geometry and the theory of boundary value problems, in particular the Dirichlet–to–Robin operator, we establish that our two–point correlation function is conformally invariant and corresponds to a boundary operator of conformal dimension one. It is plausible that the methods we use could have more general