Solutions of the Einstein Constraint Equations with Apparent Horizon Boundary

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method introduced by Isenberg for compact manifolds without boundary,suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.