We study the global geometry of solutions to Einstein's (vacuum or matter)
constraint equations of general relativity, and we establish the existence of a
broad class of asymptotically Euclidean solutions. Specifically, we associate a
solution to the Einstein equations to any given weakly asymptotically tame seed
data set satisfying suitable decay conditions, a notion we define here. Such a
data set consists of a Riemannian metric and a symmetric two-tensor prescribed
on a topological manifold with finitely many asymptotically Euclidean ends, as
well as a scalar field and a vector field describing the matter content. The
Seed-to-Solution Method we introduce here is motivated by a pioneering work by
Carlotto and Schoen on the so-called localization problem for the Einstein
equations. Our method copes with the nonlinear coupling between the Hamiltonian
and momentum constraints at the sharp level of decay, and relies on a
linearization of the Einstein equations near an arbitrary seed data set and on
estimates in a weighted Lebesgue-Holder space adapted to the problem.
Furthermore, for seed data sets enjoying stronger decay and referred to as
strongly asymptotically tame data, we prove that the seed-to-solution map (as
we call it) preserves the asymptotic behavior as well as the ADM mass of the
prescribed data. Motivated by a question raised by Carlotto and Schoen, we
define an Asymptotic Localization Problem, which we solve at the sharp level of
decay.