A well-known corollary of the positive mass theorem by Schoen-Yau is that if an asymptotically flat manifold (of non-negative scalar curvature) is exactly flat outside of a compact set, then it has to be globally flat: in other terms any such metric can never be localized inside a compact set. So what is the "optimal" localization of those metrics? For instance, can one produce scalar non-negative metrics that have positive ADM mass and still are trivial in a half-space?  In recent joint work with Schoen, we answer these questions by giving a systematic method for constructing solutions to the Einstein constraint equations that are localized inside a cone of arbitrarily small aperture. This sharply contrasts with various scalar curvature rigidity phenomena both in the closed and in the free-boundary setting. Obviously, the manifolds we get have plenty of complete (non-compact), stable minimal hypersurfaces, which is a remarkable fact since I proved that in dimension less than eight an asymptotically Schwarschild manifold containing a complete, stable minimal hypersurface has to be isometric to the Euclidean space.