Abstract: Given a conformal class of metrics on the boundary of a manifold, one can ask for the existence of an Einstein metric whose conformal infinity satisfies the boundary condition.
In 1991, Graham and Lee studied this boundary problem on the hyperbolic ball. They proved the existence of metrics sufficiently close to the round metric on a sphere by constructing approximate solutions to a quasilinear elliptic system. In his monograph (2006), Lee discussed the boundary problem on a smooth, compact manifold-with-boundary. Using a similar construction, he proved the existence and regularity results for metrics sufficiently close to a given asymptotically hyperbolic Einstein metric. The proof is based on a linear theory for Laplacian and the inverse function theorem.