我们证明了在Lorenz规范和调和规范下，爱因斯坦-杨-米尔斯系统的闵可夫斯基时空$\mathbb{R}^{1+3}$解的外部稳定性，其中杨-米尔斯场取值于与任意紧李群$G$相关的任意李代数$\cal{G}$。我们从任意充分小的初始数据出发，该数据在适合杨-米尔斯势和闵可夫斯基时空扰动的能量范数中定义，并证明了在Lorenz规范和波坐标下完全耦合的爱因斯坦-杨-米尔斯方程外部区域的柯西发展的适定性，且证明了这导致解收敛到零杨-米尔斯曲率和闵可夫斯基时空。此外，我们在波坐标下获得了Lorenz规范中杨-米尔斯势和度量的色散估计，以及杨-米尔斯曲率的规范不变范数的色散估计。这为P. Mondal和S. T. Yau的外部稳定性结果提供了一个新的证明，基于一种替代方法，使用了H. Lindblad和I. Rodnianski首次用于爱因斯坦真空方程的零标架分解。在本系列的第三篇论文中，我们详细阐述了关于我们证明的所有新材料，以便为想要学习非线性双曲微分方程和数学广义相对论中稳定性问题的博士生提供讲义。
英文摘要：
We prove the exterior stability of the Minkowski space-time, $\mathbb{R}^{1+3}$, solution to the Einstein-Yang-Mills system in both the Lorenz and harmonic gauges, where the Yang-Mills fields are valued in any arbitrary Lie algebra $\cal{G}$, associated to any compact Lie group $G$. We start with an arbitrary sufficiently small initial data, defined in a suitable energy norm for the perturbations of the Yang-Mills potential and of the Minkowski space-time, and we show the well-posedness of the Cauchy development in the exterior of the fully coupled Einstein-Yang-Mills equations in the Lorenz gauge and in wave coordinates, and we prove that this leads to solutions converging to the zero Yang-Mills curvature and to the Minkowski space-time. Furthermore, we obtain dispersive estimates in wave coordinates on the Yang-Mills potential in the Lorenz gauge and on the metric, as well as on the gauge invariant norm of the Yang-Mills curvature. This provides a new proof to the exterior stability result by P. Mondal and S. T. Yau, based on an alternative approach, by using a null frame decomposition that was first used by H. Lindblad and I. Rodnianski for the case of the Einstein vacuum equations. In this third paper of a series, we detail all the new material concerning our proof so as to provide lecture notes for Ph.D. students wanting to learn non-linear hyperbolic differential equations and stability problems in mathematical General Relativity.
