  We consider the conformal decomposition of Einstein's constraint equations
introduced by Lichnerowicz and York, on a closed manifold. We establish
existence of non-CMC weak solutions using a combination of a priori estimates
for the individual Hamiltonian and momentum constraints, barrier constructions
for the Hamiltonian constraint, and topological fixed-point arguments. An
important new feature of these results is the absense of the near-CMC
assumption when the rescaled background metric is in the positive Yamabe class,
if the freely specifiable part of the data given by the matter fields (if
present) and the traceless-transverse part of the rescaled extrinsic curvature
are taken to be sufficiently small. In this case, the mean extrinsic curvature
can be taken to be an arbitrary smooth function without restrictions on the
size of its spatial derivatives, giving what are apparently the first non-CMC
existence results without the near-CMC assumption. Standard bootstrapping
arguments to increase the regularity of the conformal factor are blocked by the
use of a weak background metric. In the CMC case, we recover Maxwell's rough
solution results as a special case. Our results extend the 1996 non-CMC result
of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed
in the case of the positive Yamabe class; (2) regularity is extended down to
the maximum allowed by the background metric and the matter; and (3) the result
holds for all three Yamabe classes. This last extension was also accomplished
recently by Allen, Clausen and Isenberg, although their result is restricted to
the near-CMC case and to smoother background metrics and data.
