Abstract. In this note we prove two existence theorems for the Einstein constraint equations on asymptotically Euclidean manifolds. The first is for arbitrary mean curvature functions with restrictions on the size of the transverse-traceless data and the non-gravitational field data, while the second assumes a near-CMC condition, with no other restrictions.

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also show that the Yamabe class of an asymptotically Euclidean manifold is the same as the Yamabe class of its conformal compactification.In particular, λ g ′ ,δ (Z) ≥ 0, and λ g ′ ,δ (Z) = 0 only if u is constant. But Z has positive measure, and therefore A(Z) does not contain any constants. Hence λ g ′ ,δ (Z) > 0, and Proposition 3.10 implies Z is Yamabe positive.This completes the proof of Theorem 4.1. Turning to the compact case (Theorem 4.2) recall that we started the AE argument with the following inessential simplifying hypotheses:1. The prescribed scalar curvature R ′ is bounded.

We continue the study of the Einstein constraint equations on compact manifolds with boundary initiated by Holst and Tsogtgerel. In particular, we consider the full system and prove existence of solutions in both the near-CMC and far-from-CMC (for Yamabe positive metrics) cases. We also make partial progress in proving the results of previous "limit equation" papers by Dahl, Gicquaud, Humbert and Sakovich.

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes.

We prove that in a certain class of conformal data on an asymptotically cylindrical manifold, if the conformally decomposed Einstein constraint equations do not admit a solution, then one can always find a nontrivial solution to the limit equation first explored by Dahl, Gicquaud, and Humbert in [DGH11]. We also give an example of a Ricci curvature condition on the manifold which precludes the existence of a solution to this limit equation, showing that such a limit criterion can be a useful tool for studying the Einstein constraint equations on manifolds with asymptotically cylindrical ends.