The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

Abstract

We establish the existence of a broad class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is prescribed arbitrarily. The proposed seed-to-solution method encompasses vacuum as well as matter spaces, and relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of(1) a Riemannian metric and a symmetric two-tensor and(2) a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame or strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (possibly with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with low decay and, simultaneously, establishing estimates that include (and go beyond) harmonic decay require significantly new arguments which are developed in the present paper. We work in a weighted Lebesgue-Hölder framework adapted to the given seed data, and we analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic regularity estimates for the linearized Einstein operator and its dual, we uncover the novel notion of mass-momentum correctors which is related to the ADM mass-momentum of the manifold. We derive precise estimates for the difference between the seed data and the associate Einstein solution —a result that should be of interest for future numerical investigations. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. By applying our seed-to-solution method with a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.