  Unique continuation results are proved for metrics with prescribed Ricci
curvature in the setting of bounded metrics on compact manifolds with boundary,
and in the setting of complete, conformally compact metrics. Related to this
issue, an isometry extension property is proved: continuous groups of
isometries at conformal infinity extend into the bulk of any complete
conformally compact Einstein metric. Relations of this property with the
invariance of the Gauss-Codazzi constraint equations under deformations are
also discussed.
