This paper is a summary of the work by the author (The global stability of the Minkowski space-time in higher dimensions, Energy estimates for the Einstein-Yang-Mills fields and applications, Exterior stability of the \((1+3)\) -dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), where we study the Einstein–Yang–Mills system in both the Lorenz and harmonic gauges, where the Yang–Mills fields are valued in any arbitrary Lie algebra \({\mathcal {G}}\) , associated with any compact Lie group G . In the paper by the author (The global stability of the Minkowski space-time in higher dimensions), we first showed that in the Lorenz gauge and in wave coordinates, we can recast the problem as equivalent to studying solutions of the Einstein–Yang–Mills equations that solve a covariant system of tensorial nonlinear hyperbolic partial differential equations. In another papers by the author (Energy estimates for the Einstein-Yang-Mills fields and applications, Exterior stability of the \({(1+3)}\) -dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), we exhibited that this system does not satisfy neither the null condition nor the weak-null condition of Lindblad–Rodnianski, rather a new weak-null condition with a different nonlinearity that has new complications that are not present neither for the Einstein vacuum equations nor for the Einstein–Maxwell system. Based on a separate energy estimate for each component, as well as based on a new separate estimate for the commutator term for the higher order energy for the tangential components, established by the author [Ghanem, S.: arXiv:2310.08611; arXiv:2310.08196], we sum up here the long detailed proof by the author (Exterior stability of the \({(1+3)}\) -dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), of the exterior stability of the Minkowski space-time, \(\mathbb {R}^{1+3}\) , governed by the fully coupled Einstein–Yang–Mills system in the Lorenz gauge, valued in any arbitrary Lie algebra \({\mathcal {G}}\) , without any assumption of spherical symmetry. This provides a first detailed proof of the exterior stability of Minkowski governed by the fully nonlinear Einstein–Yang–Mills equations in the Lorenz gauge, by using a null-frame decomposition that was first used by H. Lindblad and I. Rodnianski in their celebrated seminal work for the case of the Einstein vacuum equations. We note, however, that to the best of our knowledge, the \(L^\infty \) -estimate of Lindblad–Rodnianski does not work for the Einstein–Yang–Mills system in the Lorenz gauge. We replace their \(L^\infty \) -estimate with our new separate energy estimates for the higher-order energy norm of the tangential components that allow us to treat the new type of nonlinearity that arises from the Yang–Mills fields in the Lorenz gauge.
