Appears in collection : Nonlinear Waves Trimester - June Conference
In the presence of a translation symmetry, the 3+1 vacuum Einstein equations reduce to the 2+1 Einstein equations with a scalar fi_eld. We work in generalised wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main diffi_culty in proving global existence of solutions for small data is due to the weaker decay of free solutions to the wave equation in 2+1 dimensions, compared to 3+1 dimensions. This weak decay seems to be a deterrent for proving a stability result in the usual wave coordinates. In this talk we will present the construction of a suitable generalized wave gauge in which our system has a "cubic weak null structure",which allows for the proof of global existence.