22 April 2022
Poincaré-Einstein metrics with cusps
Asymptotically hyperbolic (or Poincaré-Einstein) Einstein metrics are important in conformal geometry and the physics of the AdS-CFT correspondence. In 1991, Graham and 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 review the proof of this result. I then discuss recent joint work with Rochon proving the existence of Einstein metrics near certain geometrically finite quotients of the hyperbolic metric.