Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the points of n 2 − N. At each λk: = n − k with k ∈ N, the resonance multiplicity 2 m(λk) and the scattering pole multiplicity ν(λk) do not always coincide: ν(λk) − m(λk) is the dimension of the kernel of a differential operator on the boundary ∂ ¯ X introduced by Graham and Zworski; in the asymptotically Einstein case, this operator is the k-th conformal Laplacian. 1.

Abstract. After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem. 1.

Abstract. The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L 2-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m, the finite time supertrace of the heat kernel on conformally compact manifolds is shown to renormalize independently of the choice of special boundary defining function.

Abstract. In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y−. 1.

Abstract. For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized Birman-Krein theory to study scattering asymptotics and, when the curvature is constant, to analyze Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the first hand, a natural spectral function ξ for the Laplacian ∆g, which replaces the counting function of the eigenvalues in this infinite volume case, and on the other hand the determinant of the scattering operator SX(λ) of ∆g on X. Both need to be defined through regularized functional: renormalized trace on the bulk X and regularized determinant on the conformal infinity ( ∂ ¯ X,[h0]). We show that det SX(λ) is meromorphic in λ, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians (Pk)k∈N of ( ∂ ¯ X,[h0]), moreover ξ(z) is proved to be the phase of det SX ( n 2 + iz) on the essential spectrum {z ∈ R+}. Applying this theory to convex cocompact quotients X = Γ\Hn+1 of hyperbolic space Hn+1, we obtain the functional equation Z(λ)/Z(n − λ) = (det SHn+1(λ)) χ(X) /det SX(λ) for Selberg zeta function Z(λ) of X, where χ(X) is the Euler characteristic of X. This describes the poles and zeros of Z(λ), computes det Pk in term of Z ( n n − k)/Z ( + k) and implies a sharp Weyl asymptotic for ξ(z). 2 2 1.

Abstract. For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of particular values of dynamical zeta functions by using natural uniformizations, one due to Mazzeo-Taylor, the other to Osgood-Phillips-Sarnak. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization. 1.

The purpose of this article is to define the radiation fields on asymptotically hyperbolic manifolds and to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was

Abstract. Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal in nity, with corresponding uniform decay on the appropriately compacti ed space. 1.

Abstract. We define for a class of even dimensional asymptotically hyperbolic manifold (X, g) a natural generalized Krein spectral function ξ (on R) with derivative the renormalized trace of the spectral measure of the Laplacian. It is related to the scattering operator S(λ) of ∆g by −2πi∂zξ(z) = TR(∂zS ( n 2 + iz)S−1 ( n − iz)) where TR is the Kontsevich-Vishik 2 trace. For even Poincaré-Einstein metrics, we define the determinant of S(λ) using methods of Kontsevich-Vishik and show that it is a conformal invariant of the conformal boundary (M,[h0]) depending meromorphically on λ, with divisors given by the resonances multiplicity and the dimensions of kernels of the conformal Laplacians (Pk)k∈N of [h0]. We finally prove that ξ is the phase of det S(λ) on the essential spectrum, we compute the determinant of Pk with respect to ξ and, as an application, det Pk is expressed explicitly in term of the Selberg zeta function for convex co-compact hyperbolic manifolds. 1.

Abstract. Inthispaperweconstructaparametrixforthehigh-energyasymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.