On an asymptotically hyperbolic manifold , Mazzeo and Melrose  have constructed the meromorphic extension of the resolvent for the Laplacian. However, there are special points on with which they did not deal. We show that the points of are at most poles of finite multiplicity and that the same property holds for the points of if and only if the metric is even. On the other hand, there exist some metrics for which has an essential singularity on , and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of approaching an essential singularity.
Colin Guillarmou. "Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds." Duke Math. J. 129 (1) 1 - 37, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12911-2