Initial-value problem of general relativity. II. Stability of solutions of the initial-value equations

We discuss the completeness and "linearization stability" of the initial-value constraints. We show that all closed solutions for which the intrinsic geometry possesses a conformal symmetry are incomplete, but stable. All closed, vacuum, moment-of-time symmetry solutions are incomplete, but only the flat case is unstable. This particular incompleteness vanishes on the addition of any source field. All other closed solutions to the initial-value constraints for which the trace of the momentum is a covariant constant are complete and stable except those solutions where the metric and the momentum have the same exact symmetry. All such closed, vacuum solutions are unstable. All asymptotically flat maximal solutions are complete and stable. In this paper we treat only the linearization stability of the initial-value constraints and make no statements about the dynamical stability of the solutions.