Geometrically Finite Asymptotically Hyperbolic Einstein metrics

Asymptotically hyperbolic Einstein metrics are important in conformal geometry and the physics of the AdS-CFT correspondence.  In 1991, Robin Graham and Jack Lee proved that any sufficiently small perturbation (in Holder norm) of the round metric on the unit sphere (with dimension at least 3) is the "conformal infinity" of an asymptotically hyperbolic Einstein metric on the open unit ball.  In this talk I will give the background and explain the proof of this result, and then discuss recent joint work with Frederic Rochon proving the existence of Einstein metrics near certain geometrically finite quotients of the hyperbolic metric.