ON HYPERBOLOIDAL CAUCHY DATA FOR VACUUM EINSTEIN EQUATIONS AND OBSTRUCTIONS TO SMOOTHNESS OF SCRI

The relationship between the geometric properties of ''hyperboloidal'' Cauchy data for vacuum Einstein equations at the conformal boundary of the initial data surface and between the space-time geometry is analyzed in detail. We prove that a necessary condition for existence of a smooth or a polyhomogeneous Scri (i.e., a Scri around which the metric is expandable in terms of r(-j) log(i) r rather than in terms of r(-j)) is the vanishing of the shear of the conformal boundary of the initial data surface. We derive the ''boundary constraints'' which have to be satisfied by an initial data set for compatibility with Friedfich's conformal framework. We show that a sufficient condition for existence of a smooth Scri (not necessarily complete) is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data. We also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non-vanishing of the Weyl tensor at the conformal boundary.