  The 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.
