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General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG
Abstract:
In the Cauchy problem for asymptotically flat vacuum data the solution-jets
along the cylinder at space-like infinity develop in general logarithmic
singularities at the critical sets at which the cylinder touches future/past
null infinity. The tendency of these singularities to spread along the null
generators of null infinity obstructs the development of a smooth conformal
structure at null infinity. For the solution-jets arising from time reflection
symmetric data to extend smoothly to the critical sets it is necessary that the
Cotton tensor of the initial three-metric h satisfies a certain conformally
invariant condition (*) at space-like infinity, it is sufficient that h be
asymptotically static at space-like infinity. The purpose of this article is to
characterize the gap between these conditions. We show that with the class of
metrics which satisfy condition (*) on the Cotton tensor and a certain
non-degeneracy requirement is associated a one-form $\kappa$ with conformally
invariant differential $d\kappa$. We provide two criteria: If $h$ is real
analytic, $\kappa$ is closed, and one of it integrals satisfies a certain
equation then h is conformal to static data near space-like infinity. If h is
smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a
certain equation asymptotically then h is asymptotically conformal to static
data at space-like infinity.
