Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Abstract

On an asymptotically hyperbolic manifold $\left(X^{n+1},g\right)$, Mazzeo and Melrose [18] have constructed the meromorphic extension of the resolvent $R\left(\lambda \right):={\left({\Delta }_g-\lambda \left(n-\lambda \right)\right)}^{-1}$ for the Laplacian. However, there are special points on $\left(1/2\right)\left(n-\mathbb{N}\right)$ with which they did not deal. We show that the points of $\left(n/2\right)-\mathbb{N}$ are at most poles of finite multiplicity and that the same property holds for the points of $\left(\left(n+1\right)/2\right)-\mathbb{N}$ if and only if the metric is even. On the other hand, there exist some metrics for which $R\left(\lambda \right)$ has an essential singularity on $\left(\left(n+1\right)/2\right)-\mathbb{N}$, and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of $R\left(\lambda \right)$ approaching an essential singularity.