Stability of asymptotic behaviour within polarized [Formula: see text]-symmetric vacuum solutions with cosmological constant.

We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized [Formula: see text]-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant [Formula: see text]. This stability result generalizes the results proven in Ames E (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. .

Gluing initial data sets for general relativity.

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural.