  This paper studies several aspects of asymptotically hyperbolic Einstein
metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for
such metrics and establish uniqueness under natural conditions on the boundary
data. By examination of explicit black hole metrics, it is shown that neither
uniqueness nor finiteness holds in general for AH Einstein metrics with a
prescribed conformal infinity. We then describe natural conditions which are
sufficient to ensure finiteness.
