Abstract: We study regularity near infinity of asymptotically flat solutions to the Einstein vacuum equations in n+1 dimensions. We prove that, when n=3 and n=4,6,...,2k, the metric on the quotient space of the timelike Killing vector is conformally analytic at the point-at-infinity. The proof of this result for n=3,4 makes important use of the conformally invariant tensors in these domensions (Cotton tensor for n=3, Weyl and Bach tensor for n=4). The proof for general even n should involve the Fefferman-Graham obstruction tensors (generalizing the Bach tensor). Unfortunately we are unable to follow this route, but use a very non-geometrical argument which only works for n=6,8,...,2k.

Reference: R.Beig, P.T.Chrusciel arXiv:gr-qc/0612012