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The conformal method developed in the 1970s and the more recent Lagrangianand Hamiltonian conformal thin-sandwich methods are techniques for findingsolutions of the Einstein constraint equations. We show that they aremanifestations of a single conformal method: there is a straightforward way toconvert back and forth between the parameters for these methods so that thecorresponding solutions of the Einstein constraint equations agree. Theunifying idea is the need to clearly distinguish tangent and cotangent vectorsto the space of conformal classes on a manifold, and we introduce a vocabularyfor working with these objects without reference to a particular representativebackground metric. As a consequence of these conceptual advantages, wedemonstrate how to strengthen previous near-CMC existence and non-existencetheorems for the original conformal method to include metrics with scalarcurvatures that change sign.

The Conformal Method and the Conformal Thin-Sandwich Method Are the Same