Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on $Z$, and are equivalent to the usual constraint equations that $Z$ satisfies as a spacelike hypersurface in a spacetime satisfying Einstein's vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the `classical' method of Lichnerowicz and York that is used to solve the usual constraint equations.