  We consider asymptotically Euclidean, initial data sets for Einstein's field
equations and solve the localization problem at infinity, also called gluing
problem. We achieve optimal gluing and optimal decay, in the sense that we
encompass solutions with possibly arbitrarily low decay at infinity and
establish (super-)harmonic estimates within possibly arbitrarily narrow conical
domains. In the localized seed-to-solution method (as we call it), we define a
variational projection operator which associates the solution to the Einstein
constraints that is closest to any given localized seed data set (as we call
it). Our main contribution concerns the derivation of harmonic estimates for
the linearized Einstein operator and its formal adjoint which, in particular,
includes new analysis on the linearized scalar curvature operator. The
statement of harmonic estimates requires the notion of energy-momentum
modulators (as we call them), which arise as correctors to the localized seed
data sets. For the Hamiltonian and momentum operators, we introduce a notion of
harmonic-spherical decomposition and we uncover stability conditions on the
localization function, which are localized Poincare and Hardy-type inequalities
and, for instance, hold for arbitrarily narrow gluing domains. Our localized
seed-to-solution method builds upon the gluing techniques pioneered by
Carlotto, Chrusciel, Corvino, Delay, Isenberg, Maxwell, and Schoen, while
providing a proof of a conjecture by Carlotto and Schoen on the localization
problem and generalize P. LeFloch and Nguyen's theorem on the asymptotic
localization problem.
