Elliptic-hyperbolic systems and the Einstein equations

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 quasi-linear 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 space-times.