  In this work, we use the global analysis and degree-theoretic methods
introduced by Smale to study the existence and multiplicity of solutions of the
vacuum Einstein constraint equations given by the conformal method of
Lichnerowicz-Choquet-Bruhat-York. In particular this approach gives a new proof
of the existence result of Maxwell and Holst-Nagy-Tsogtgerel. We also relate
the method to the limit equation of Dahl-Gicquaud-Humbert and the non-existence
result of Nguyen.
