Constraint Equations for 3 + 1 Vacuum Einstein Equations with a Translational Space-Like Killing Field in the Asymptotically Flat Case

We solve the Einstein constraint equations for a 3 + 1- dimensional vacuum space–time with a space-like translational Killing field. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1-dimensional problem to a 2 + 1-dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history. In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on $${\mathbb{R}^2}$$R2. The main difficulty in that case is due to the delicate inversion of the Laplacian on $${\mathbb{R}^2}$$R2. In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.