Prescribed Ricci curvature near an Einstein manifold with boundary

Let ( M , g ) (M,g) be a compact Einstein Riemannian manifold with boundary. We show that under certain conditions, the map that associates to a metric on M M its Ricci curvature, its induced conformal class on the boundary, and its mean curvature on the boundary is locally invertible near g g . The contravariant Ricci operator as well as other operators such as the Einstein operator are also studied.