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Weakly asymptotically hyperbolic solutions to the Einstein constraint equations Allen, Paul T


We introduce a class of ``weakly asymptotically hyperbolic'' geometries whose sectional curvatures tend to \(-1\) and are \(C^0\) but not necessarily \(C^2\) conformally compact. We establish Fredholm results for geometric elliptic operators in this setting. We use these results to construct constant-mean-curvature solutions to the Einstein constraint equations in the weakly asymptotically hyperbolic setting. We furthermore can ensure that our solutions satisfy the shear-free condition, which is necessary for any spacetime development to admit a regular conformal boundary at future null infinity. This is joint work with James Isenberg, John M. Lee, and Iva Stavrov Allen.

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