  The conformal method is a technique for finding Cauchy data in general
relativity solving the Einstein constraint equations, and its parameters
include a conformal class, a conformal momentum (as measured by a densitized
lapse), and a mean curvature. Although the conformal method is successful in
generating constant mean curvature (CMC) solutions of the constraint equations,
it is unknown how well it applies in the non-CMC setting, and there have been
indications that it encounters difficulties there. We are therefore motivated
to investigate alternative generalizations of the CMC conformal method.
  Introducing a densitized lapse into the ADM Lagrangian, we find that
solutions of the momentum constraint can be described in terms of three
parameters. The first is conformal momentum as it appears in the standard
conformal method. The second is volumetric momentum, which appears as an
explicit parameter in the CMC conformal method, but not in the non-CMC
formulation. We have called the third parameter drift momentum, and it is the
conjugate momentum to infinitesimal motions in superspace that preserve
conformal class and volume form up to independent diffeomorphisms. This
decomposition of solutions of the momentum constraint leads to extensions of
the CMC conformal method where conformal and volumetric momenta both appear as
parameters. There is more than one way to treat drift momentum, in part because
of an interesting duality that emerges, and we identify three candidates for
incorporating drift into a variation of the conformal method.
