Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Abstract

On an asymptotically hyperbolic manifold $(X^{n+1},g)$, Mazzeo and Melrose [18] 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 $({1}/{2})(n-\mathbb{N})$ with which they did not deal. We show that the points of $({n}/{2})-\mathbb{N}$ are at most poles of finite multiplicity and that the same property holds for the points of $(({n+1})/{2})-\mathbb{N}$ 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 $(({n+1})/{2})-\mathbb{N}$, and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of $R(\lambda)$ approaching an essential singularity.