The wave trace on asymptotically complex hyperbolic manifolds

The behavior of solutions to the wave equation on a Riemannian manifold has a close relationship with both the behavior of geodesics and the spectrum of the Laplacian. In this talk, following Duistermaat-Guillemin, we study the wave solution operator on asymptotically complex hyperbolic manifolds and analyze of its (regularized) trace. On this class of non-compact manifolds, modeled on the complex ball with its Bergman metric, we prove in particular that the leading term arises as a renormalized volume of this non-compact space, and prove a relationship between singularities of the wave trace and the lengths of closed geodesics.