Upper and Lower Bounds on Resonances for Manifolds Hyperbolic Near Infinity

Abstract

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r n+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an r n+1 lower bound on the counting function for scattering poles.