Exterior stability of the Minkowski space-time governed by the Einstein-Yang-Mills equations in the Lorenz gauge

I shall start by presenting the Einstein-Yang-Mills system and by writing it in the Lorenz gauge and in wave coordinates as a coupled system of non-linear hyperbolic partial differential equations. This system of partial differential equations has new interesting source terms which were not treated previously in the literature and which present their own complications. Based on this hyperbolic formulation, I will present the idea behind the proof of the non-linear stability of the Minkowski space-time, solution to the Einstein-Yang-Mills equations, in the Lorenz gauge and in wave coordinates, in all space dimensions greater or equal to three, based on a continuity argument for a higher order weighted energy norm. In the critical case of three space-dimensions, we use a null frame decomposition, that was first used by Lindblad and Rodnianski for the Einstein vacuum equations. We then deal with new difficulties that do not exist for Einstein vacuum nor for Einstein-Maxwell fields. In particular, we treat new terms that have a different structure in the non-linearities, which present their own complications, and we derive more refined energy estimates and a more refined formula to estimate the commutator term.