Existence of stationary vacuum solutions of Einstein's equations in an exterior domain

A proof is given for the existence and uniqueness of a stationary vacuum solution (ℳ, g, ξ) of the boundary value problem consisting of Einstein's equations in an exterior domain ℳ diffeomorphic to ℝ x Σ (where Σ = ℝ3\B(0, R)) and boundary data depending on the Killing field ξ on ∂Σ. The boundary data must be sufficiently close to that of a stationary, spatially conformally flat vacuum solution (ℳ, g̊, ξ̊). © Australian Mathematical Society 1999.