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In this article we continue our effort to do a systematic development of thesolution theory for conformal formulations of the Einstein constraint equationson compact manifolds with boundary. By building in a natural way on our recentwork in Holst and Tsogtgerel (2013), and Holst, Nagy, and Tsogtgerel (2008,2009), and also on the work of Maxwell (2004, 2005, 2009) and Dain (2004),under reasonable assumptions on the data we prove existence of both near- andfar-from-constant mean curvature solutions for a class of Robin boundaryconditions commonly used in the literature for modeling black holes, with athird existence result for constant mean curvature (CMC) appearing as a specialcase. Dain and Maxwell addressed initial data engineering for space-times thatevolve to contain black holes, determining solutions to the conformalformulation on an asymptotically Euclidean manifold in the CMC setting, withinterior boundary conditions representing excised interior black hole regions.Holst and Tsogtgerel compiled the interior boundary results covered by Dain andMaxwell, and then developed general interior conditions to model the apparenthorizon boundary conditions of Dain and Maxwell for compact manifolds withboundary, and subsequently proved existence of solutions to the Lichnerowiczequation on compact manifolds with such boundary conditions. This paper picksup where Holst and Tsogtgerel left off, addressing the general non-CMC case forcompact manifolds with boundary. As in our previous articles, our focus here isagain on low regularity data and on the interaction between different types ofboundary conditions. While our work here serves primarily to extend thesolution theory for the compact with boundary case, we also develop severaltechnical tools that have potential for use with the asymptotically Euclideancase.

Non-CMC Solutions of the Einstein Constraint Equations on Compact Manifolds with Apparent Horizon Boundaries