  In this dissertation, we prove a number of results regarding the conformal
method of finding solutions to the Einstein constraint equations. These results
include necessary and sufficient conditions for the Lichnerowicz equation to
have solutions, global supersolutions which guarantee solutions to the
conformal constraint equations for near-constant-mean-curvature (near-CMC) data
as well as for far-from-CMC data, a proof of the limit equation criterion in
the near-CMC case, as well as a model problem on the relationship between the
asymptotic constants of solutions and the ADM mass. We also prove a
characterization of the Yamabe classes on asymptotically Euclidean manifolds
and resolve the (conformally) prescribed scalar curvature problem on
asymptotically Euclidean manifolds for the case of nonpositive scalar
curvatures.
  Many, though not all, of the results in this dissertation have been
previously published in [Dilts13b], [DIMM14], [DL14], [DM15], and [DGI15]. This
article is the author's Ph.D. dissertation, except for a few minor changes.
