Stability in exponential time of Minkowski space-time with a space-like translation symmetry

In this note, we discuss the nonlinear stability in exponential time of Minkowki space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the 3+1 vacuum Einstein equations reduce to the 2+1 Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in 1 √ t of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully chose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity. This article appears under the same form in the proceedings of the Laurent Schwartz seminar.