  In this article we consider the conformal decomposition of the Einstein
constraint equations introduced by Lichnerowicz, Choquet-Bruhat, and York, on
asymptotically flat (AF) manifolds. Using the non-CMC fixed-point framework
developed in 2009 by Holst, Nagy, and Tsogtgerel and by Maxwell, we establish
existence of coupled non-CMC weak solutions for AF manifolds. As is the case
for the analogous existence results for non-CMC solutions on closed manifolds
and compact manifolds with boundary, our results here avoid the near-CMC
assumption by assuming that the freely specifiable part of the data given by
the traceless-transverse part of the rescaled extrinsic curvature and the
matter fields are sufficiently small. The non-CMC rough solutions results here
for AF manifolds may be viewed as extending to AF manifolds the 2009 and 2014
results on rough far-from-CMC positive Yamabe solutions for closed and compact
manifolds with boundary. Similarly, our results may be viewed as extending the
recent 2014 results for AF manifolds of Dilts, Isenberg, Mazzeo and Meier, and
of Holst and Meier; while their results are restricted to smoother background
metrics and data, the results here allow the regularity to be extended down to
the minimum regularity allowed by the background metric and the matter, further
completing the rough solution program initiated by Maxwell and Choquet-Bruhat
in 2004.
