  The conformal method developed in the 1970s and the more recent Lagrangian
and Hamiltonian conformal thin-sandwich methods are techniques for finding
solutions of the Einstein constraint equations. We show that they are
manifestations of a single conformal method: there is a straightforward way to
convert back and forth between the parameters for these methods so that the
corresponding solutions of the Einstein constraint equations agree. The
unifying idea is the need to clearly distinguish tangent and cotangent vectors
to the space of conformal classes on a manifold, and we introduce a vocabulary
for working with these objects without reference to a particular representative
background metric. As a consequence of these conceptual advantages, we
demonstrate how to strengthen previous near-CMC existence and non-existence
theorems for the original conformal method to include metrics with scalar
curvatures that change sign.
