Journal of Differential Geometry

On the Asymptotics for the Vacuum Einstein Constraint Equations

Justin Corvino and Richard M. Schoen

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In this paper we prove density of asymptotically flat solutions with special asymptotics in general classes of solutions of the vacuum constraint equations. The first type of special asymptotic form we consider is called harmonic asymptotics. This generalizes in a natural way the conformally flat asymptotics for the K = 0 constraint equations. We show that solutions with harmonic asymptotics form a dense subset (in a suitable weighted Sobolev topology) of the full set of solutions. An important feature of this construction is that the approximation allows large changes in the angular momentum. The second density theorem we prove allows us to approximate asymptotically flat initial data on a three-manifold M for the vacuum Einstein field equation by solutions which agree with the original data inside a given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The construction generalizes work in [C], where the time-symmetric (K = 0) case was studied.

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J. Differential Geom., Volume 73, Number 2 (2006), 185-217.

First available in Project Euclid: 27 April 2006

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Corvino, Justin; Schoen, Richard M. On the Asymptotics for the Vacuum Einstein Constraint Equations. J. Differential Geom. 73 (2006), no. 2, 185--217. doi:10.4310/jdg/1146169910.

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