We study the local well-posedness of the Einstein-Yang-Mills equations in constant mean extrinsic curvature spatial harmonic generalized Coulomb gauge. In this choice of gauge, the complete Einstein-Yang-Mills equations reduce to a coupled elliptic-hyperbolic system. We establish the existence of a unique, local, continuous-in-time solution to this coupled system. This yields an “in time” continuation criteria of the solutions which are to be used in the potential future proof of improved continuation criteria for this coupled system utilizing Moncrief’s light cone estimate technique.

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