On the global well-posedness of the Einstein–Yang–Mills system

Abstract

In this paper, we present a partial result on the global well-posedness of the Cauchy problem for the Einstein–Yang–Mills system in constant mean extrinsic curvature spatial harmonic and generalized Coulomb gauges as introduced in the work of Mondal [arXiv:2112.14273 (2021)]. We give a small-data global existence theorem for a family of n + 1 dimensional spacetimes with n ≥ 4, utilizing energy arguments presented in the work of Andersson and Moncrief [J. Differ. Geom. 89, 1–47 (2009)]. We observe that these energy arguments will fail for n = 3 due to the conformal invariance of 3 + 1 Yang–Mills equations and present a gauge-covariant formulation of the Einstein–Yang–Mills system in 3 + 1 dimensions to show that an energy argument cannot be used to prove the global well-posedness result, regardless of the choice of gauge.