A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method

Let ( M , g ) be a compact Riemannian manifold on which a trace-free and divergence-free σ ∈ W 1 , p and a positive function τ ∈ W 1 , p , p > n are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data σ and τ . We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on 1 -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 g for the C 0 -topology.