  We construct and parametrize solutions to the constraint equations of general
relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed
decay properties at infinity. We thus provide a large class of initial data for
the results on stability of Minkowski which include a mass term in the
asymptotics. Due to the symmetries of Minkowski, a naive linear perturbation
fails. Our construction is based on a simplified conformal method, a reduction
to transverse traceless perturbations and a nonlinear fixed point argument
where we face linear obstructions coming from the cokernels of both the
linearized constraint operator and the Laplace operator. To tackle these
obstructions, we introduce a well-chosen truncated black hole around which to
perturb. The control of the parameters of the truncated black hole is the most
technical part of the proof, since its center of mass and angular momentum
could be arbitrarily large.
