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

TL;DR

This paper investigates the vacuum Einstein constraint equations via the conformal method, establishing a limit equation that characterizes solvability and providing conditions for existence of solutions on generic metrics.

Contribution

It introduces a limit equation linked to the solvability of the Einstein constraints and derives explicit conditions ensuring solutions exist for a dense set of metrics.

Findings

Non-existence of solutions implies a non-trivial solution to the limit equation.

Conditions are identified under which the limit equation has no solutions.

Existence of solutions is guaranteed under explicit assumptions on the data.

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 using the well known conformal method with data $\sigma$ and $\tau$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get 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.