Let be a compact Riemannian manifold on which a trace-free and divergence-free and a positive function , are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data and . We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on -forms. This last equation can be shown to be without solutions in many situations. As a corollary, we get the existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which, in particular, hold on a dense set of metrics for the -topology.
Mattias Dahl. Romain Gicquaud. Emmanuel Humbert. "A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method." Duke Math. J. 161 (14) 2669 - 2697, 1 November 2012. https://doi.org/10.1215/00127094-1813182