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Conformal structures of static vacuum data

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 $κ$ with conformally invariant differential $dκ$. We provide two criteria: If $h$ is real analytic, $κ$ 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, $κ$ 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.