Stability of Minkowski Space-time with a translation space-like Killing eld

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing eld. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar eld. 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 diculty in this paper 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. 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. 1 Introduction In this paper, we address the stability of the Minkowski solution to the Einstein vacuum equations with a translation space-like Killing eld. More precisely, we look for solutions of the 3 + 1 vacuum Einstein equations, on manifolds of the form Σ × R x 3 × R t , where Σ is a 2 dimensional manifold, equipped with a metric of the form g = e −2φ g + e 2φ (dx 3) 2 , where φ is a scalar function, and g a Lorentzian metric on Σ × R, all quantities being independent of x 3. For these metrics, Einstein vacuum equations are equivalent to the 2 + 1 dimensional system g φ = 0 R µν = 2∂ µ φ∂ ν φ, (1.1) where R µν is the Ricci tensor associated to g. Choquet-Bruhat and Moncrief studied the case where Σ is compact of genus G ≥ 2. In [7], they proved the stability of a particular expanding solution. In this paper we work in the case Σ = R 2. A particular solution is then given by Minkowski solution itself. It corresponds to φ = 0 and g = m, the Minkowski metric in dimension 2 + 1. A natural question one can ask in this setting is the nonlinear stability of this solution. In the 3+1 vacuum case, the stability of Minkowski space-time has been proven in the celebrated work of Christodoulou and Klainerman [8] in a maximal foliation. It has then been proven by Lindblad and Rodnianski using wave coordinates in [23]. Their proof extends also to Einstein equations coupled to a scalar eld. Let us note that the perturbations of Minkowski solution considered in our paper are not asymptotically at in 3 + 1 dimension, due to the presence of a translation Killing eld. Consequently they are not included in [8] and [23]