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Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    David Maxwell

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  1. David Maxwell
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Communicated by G.W. Gibbons

Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the paper’s style. This research was partially supported by NSF grant DMS-0305048.

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Maxwell, D. Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries. Commun. Math. Phys. 253, 561–583 (2005). https://doi.org/10.1007/s00220-004-1237-x

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