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In this paper, we prove the nonlinear stability in exponential time ofMinkowki space-time with a translation space-like Killing field. In thepresence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the2 + 1 Einstein equations with a scalar field. We work in generalised wavecoordinates. In this gauge Einstein equations can be written as a system ofquasilinear quadratic wave equations. The main difficulty in this paper is dueto the decay in $\frac{1}{\sqrt{t}}$ of free solutions to the wave equation in2 dimensions, which is weaker than in 3 dimensions. We have to rely on theparticular structure of Einstein equations in wave coordinates. We also have tocarefully choose the behaviour of our metric in the exterior region to enforceconvergence to Minkowski space-time at time-like infinity.

Stability in exponential time of Minkowski Space-time with a translation space-like Killing field