Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Abstract

On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ) := (Δg - λ(n - λ))-1 for the Laplacian. However, there are special points on (1/2)(n - 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((n + 1)/2) - ℕ if and only if the metric is even. On the other hand, there exist some metrics for which R(λ) 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(λ) approaching an essential singularity.