### Abstract

We show that the space of solutions to the (Riemannian)Einstein equations on a bounded domain is either empty or an infinite dimensional Banach manifold for which the map to the metric on the boundary is Fredholm, of index 0. The same result holds for metrics with compact "inner" boundary with (for instance) asymptotically flat ends. It also holds for the Einstein equations coupled to general matter fields, and in all dimensions. Applied to the static (or stationary) vacuum Einstein equations, the result is relevant to Bartnik's static extension conjecture, and generalizes results of P. Miao.