Refined gluing for vacuum Einstein constraint equations

Wefirstshowthattheconnectedsumalongsubmanifoldsintroducedbythesecond author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in every case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.