Understand
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.