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On the asymptotics for the vacuum Einstein constraint equations. (English) Zbl 1122.58016

The authors prove the 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 under consideration is called harmonic asymptotics. This generalizes in a natural way the conformally flat asymptotics for the \(K=0\) constraint equations. The authors show that the solutions with harmonic asymptotics form a dense subset 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 question in examination is the approximation of the asymptotically flat initial data on a three manifold \(M\) for the vacuum Einstein equations, 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 [J. Corvino, Commun. Math. Phys. 214, No. 1, 137–189 (2000; Zbl 1031.53064)], where the time-symmetric \((K=0)\) case was studied.

MSC:

58J45 Hyperbolic equations on manifolds
35Q75 PDEs in connection with relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)

Citations:

Zbl 1031.53064
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