  For each set of (freely chosen) seed data, the conformal method reduces the
Einstein constraint equations to a system of elliptic equations, the conformal
constraint equations. We prove an admissibility criterion, based on a
(conformal) prescribed scalar curvature problem, which provides a necessary
condition on the seed data for the conformal constraint equations to (possibly)
admit a solution. We then consider sets of asymptotically Euclidean (AE) seed
data for which solutions of the conformal constraint equations exist, and
examine the blowup properties of these solutions as the seed data sets approach
sets for which no solutions exist. We also prove that there are AE seed data
sets which include a Yamabe nonpositive metric and lead to solutions of the
conformal constraints. These data sets allow the mean curvature function to
have zeroes.
