Stability of Minkowski Space-Time with a Translation Space-Like Killing Field

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing field. 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 generalised wave coordinates. In this gauge Einstein’s equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in this paper is due to the decay in \(\frac{1}{\sqrt{t}}\) of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. This weak decay seems to be an obstruction for proving a stability result in the usual wave coordinates. In this paper we construct a suitable generalized wave gauge in which our system has a “cubic weak null structure”, which allows for the proof of global existence.