Initial Data for Black Holes and Rough Spacetimes

I'll describe the construction of two new classes of solutions of the Einstein constraint equations. The first construction generates solutions on asymptotically Euclidean manifolds with boundary such that the boundary is an apparent horizon. This initial data has application to the general relativistic $N$-body problem. The second construction yields a family of low regularity asymptotically Euclidean solutions of the constraint equations. These solutions have a metric in $H^s\loc$ with $s>3/2$ and are required for answering well-posedness questions for the Einstein evolution equations. In both cases, the classical conformal method of solving the constraint equations proves to be sufficiently robust to extend to these new settings.