Resonances for Obstacles in Hyperbolic Space

We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound Im λ ≤ - 1 2 , which is optimal in dimension 2. In odd dimensions we also show that Im λ ≤ - μ ρ for a universal constant μ , where ρ is the radius of a ball containing the obstacle; this gives an improvement for small obstacles. In dimensions 3 and higher the proofs follow the classical vector field approach of Morawetz, while in dimension 2 we obtain our bound by working with spaces coming from general relativity. We also show that in odd dimensions resonances of small obstacles are close, in a suitable sense, to Euclidean resonances.