On constructing asymptotically flat initial data using the evolutionary formulation of the constraints*

Since the introduction of the evolutionary forms of the vacuum Einstein constraints, it has been a great challenge and desire to be able to construct asymptotically flat initial data sets. Here we introduce a method that, by controlling only the monopole part of a single freely specifiable variable, guarantees the asymptotic flatness of solutions to either the parabolic-hyperbolic or algebraic-hyperbolic formulations. Furthermore, our method includes a tuning parameter that allows the production of either strongly asymptotically flat or weakly asymptotically flat initial data with prescribed fall-off rates. To support our claims, we construct near Kerr initial data with various fall-off rates by integrating the evolutionary forms of the constraints numerically.