Initial data for stationary spacetimes near spacelike infinity

We study Cauchy initial data for asymptotically flat, stationary vacuum spacetimes near spacelike infinity. The fall-off behaviour of the intrinsic metric and the extrinsic curvature is characterized. We prove that they have an analytic expansion in powers of a radial coordinate. The coefficients of the expansion are analytic functions of the angles. This result allow us to fill a gap in the proof found in the literature of the statement that all asymptotically flat, vacuum stationary spacetimes admit an analytic compactification at null infinity. Stationary initial data are physically important and highly non-trivial examples of a large class of data with similar regularity properties at spacelike infinity, namely, initial data for which the metric and the extrinsic curvature have asymptotic expansion in terms of powers of a radial coordinate. We isolate the property of the stationary data which is responsible for this kind of expansion.