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

Abstract

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p>n$ are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data $\sigma$ and $\tau$. 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.