  It is shown how the extended conformal Einstein field equations and a gauge
based on the properties of conformal geodesics can be used to analyse the
non-linear stability of Einstein spaces with negative scalar curvature. This
class of spacetimes admits a smooth conformal extension with a spacelike
conformal boundary. Central to the analysis is the use of conformal Gaussian
systems to obtain a hyperbolic reduction of the conformal Einstein field
equations for which standard Cauchy stability results for symmetric hyperbolic
systems can be employed. The use of conformal methods allows to rephrase the
question of global existence of solutions to the Einstein field equations into
considerations of finite existence time for the conformal evolution system.
