In 2014, Carlotto and Schoen constructed initial data sets that solve the vacuum Einstein constraints and that interpolate between any asymptotically-flat vacuum solution in a cone and Euclidean space outside a wider cone. Starting from a naive interpolation (g,K) of the two solutions to be glued, they corrected it to an exact solution that is asymptotic flat with a power-law decay slightly worse than that of (g,K).
With Philippe G. LeFloch, we reached an optimal version of their gravitational shielding by proving estimates whose power-law decay is controlled by the accuracy with which (g,K) solves the constraints, even beyond harmonic decay (namely the decay rate of black hole metrics). At the harmonic decay rate, we encounter corrections arising from a silhouette function and vector, as we call them, in the kernel of asymptotic operators built from the linearized constraints. Our work allows for very slow decay of the metrics, in which case one must define the relative ADM energy and momentum of a pair of sufficiently close initial data sets.
