2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/34030The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well--posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum spacetimes.29 pages, several minor corrections, submitted to Ann. H. PoincareGeneral Relativity and Quantum CosmologyAnalysis of PDEstext