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 o f the initial data surface and between the space-time geometry is anal yzed in detail. We prove that a necessary condition for existence of a smooth or a polyhomogeneous Scri (i.e., a Scri around which the metri c 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 ini tial data surface. We derive the ''boundary constraints'' which have t o be satisfied by an initial data set for compatibility with Friedfich 's conformal framework. We show that a sufficient condition for existe nce of a smooth Scri (not necessarily complete) is the vanishing of th e shear of the conformal boundary of the initial data surface and smoo thness up to boundary of the conformally rescaled initial data. We als o show that the occurrence of some log terms in an asymptotic expansio n at the conformal boundary of solutions of the constraint equations i s related to the non-vanishing of the Weyl tensor at the conformal bou ndary.