Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose [ 181 have constructed the meromorphic extension of the resolvent R(lambda) := (Delta(g) - lambda (n - lambda))(-1) for the Laplacian. However, there are special points on (1/2)(1) - N) with which they did not deal. We show that the points of (n/2) - N are at most poles of finite multiplicity and that the same property holds for the points of ((it + 1)/2) - N if and only if the metric is even. On the other hand, there exist some metrics for which R(lambda) has an essential singularity on ((n + 1)/2)-N, and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R (A) approaching an essential singularity.