We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in H s loc with s > 3=2. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations.Brought to you by | Freie Universität Berlin Authenticated Download Date | 7/8/15 6:55 AM We recall that in the classical setting, the conformal method of Lichnerowicz [18], Choquet-Bruhat and York [11] provides a parameterization of all maximal (i.e. tr K ¼ 0), asymptotically Euclidean (AE) solutions. We now review this method, as it also forms the basis of our construction of rough solutions. Hereafter we suppose M is an n-manifold with n f 3. We seek a solution of the form 2 Maxwell, Rough solutions of the Einstein constraint equations Brought to you by |

Abstract. We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in H s where s > 3/2. The constant mean curvature (CMC) conformal method leads to a construction of all CMC initial data with this level of regularity. These results extend a construction from [Ma04] that treated the asymptotically Euclidean case.