The seed-to-solution method for the Einstein equations

Philippe G. LeFloch, The-Cang Nguyen

Published 2019-03-01Version 1

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.