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. . (doi:10.1007/s00023-021-01142-0)), which focus on the [Formula: see text] case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for [Formula: see text], the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized [Formula: see text]-symmetric vacuum solutions than those considered in Ames E (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. . (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G (2020 . Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in [Formula: see text] for some [Formula: see text], for certain families of polarized [Formula: see text]-symmetric solutions with cosmological constant. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.