The global stability of the Minkowski space-time solution to the Einstein-Yang-Mills equations in higher dimensions

Abstract

This is a first in a series of papers in which we study the stability of the $(1+n)$-Minkowski space-time, for $n \geq 3$, solution to the Einstein-Yang-Mills equations, in both the Lorenz and harmonic gauges, associated to any arbitrary compact Lie group $G$, and for arbitrary small perturbations. In this first, we prove global stability of the Minkowski space-time, $\mathbb{R}^{1+n}$, in higher dimensions $n \geq 5$ (both in the interior and in the exterior); in the paper that follows, we prove exterior stability for $n=4$; and its sequel, we prove exterior stability for $n=3$, and in all these cases, stability is studied as a solution to the fully coupled Einstein-Yang-Mills system in the Lorenz and harmonic gauges. We show here that for $n \geq 5$, the $\mathbb{R}^{1+n}$ Minkowski space-time in wave coordinates is stable as solution to the Einstein-Yang-Mills system in the Lorenz gauge on the Yang-Mills potential, for sufficiently small perturbations of the Einstein-Yang-Mills potential and metric, and leads to a global Cauchy development. We also obtain dispersive estimates in wave coordinates on the gauge invariant norm of the Yang-Mills curvature, on the Yang-Mills potential in the Lorenz gauge, and on the perturbations of the metric. In this manuscript, we detail all the material of our proof so as to provide lecture notes for Ph.D. students wanting to learn the Cauchy problem for the Einstein-Yang-Mills system.