Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

TL;DR

This paper develops methods to construct solutions to the Einstein constraint equations with apparent horizon boundaries, extending existing techniques to manifolds with boundary and improving the theory for asymptotically Euclidean spaces.

Contribution

It introduces sufficient conditions for the conformal method to produce solutions with apparent horizon boundaries, extending the barrier method to boundary cases and low regularity metrics.

Findings

Constructed asymptotically Euclidean solutions with apparent horizon boundaries.

Extended the barrier method to boundary conditions and low regularity metrics.

Improved the understanding of Einstein constraint equations on manifolds with boundary.

Abstract

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.