Sharp upper bounds on resonances for perturbations of hyperbolic space

For certain compactly supported metric and/or potential perturbations of the Laplacian on ${\mathbb{H}}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in ${\mathbb{H}}^{n+1}$, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.