Sharp upper bounds on resonances for perturbations of hyperbolic space

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For certain compactly supported metric and/or potential perturbations of the Laplacian on Hⁿ⁺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 Hⁿ⁺¹, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.