  We adapt the Bartnik method to provide a Hilbert manifold structure for the
space of solutions, without KID's, to the vacuum constraint equations on
compact manifold of any dimension $\geq 3$. In the course, we prove that some
fibers of the scalar curvature or the constraint operator are Hilbert
submanifolds. We also study some operators and inequalities related to the
KID's operator. Finally we comment the adaptation to some non compact
manifolds.
