Perturbations of Fefferman spaces over CR three-manifolds

Abstract

We introduce a generalisation of Fefferman's conformal circle bundle over a contact Cauchy-Riemann three-manifold. These can be viewed as exact `perturbations' of Fefferman's structure by a semi-basic one-form, which encodes additional data on the CR three-manifold. We find conditions on the Weyl tensor and the Bach tensor for a Lorentzian conformal four-manifold equipped with a twisting non-shearing congruence of null geodesics to be locally conformally isometric to such a perturbed Fefferman space. We investigate the existence of metrics in the perturbed Fefferman conformal class satisfying appropriate sub-conditions of the Einstein equations, such as the so-called pure radiation equations. These metrics are defined only off cross-sections of Fefferman's circle bundle, and are conveniently expressed in terms of densities that generalise Gover's notion of almost Einstein scales. Our setup allows us to reduce the prescriptions on the Ricci tensor to the underlying CR three-manifold in terms of differential constraints on a complex density and the CR data of the perturbation one-form. One such constraint turns out to arise from a non-linear, or gauged, analogue of a second-order differential operator on densities. A solution thereof provides a criterion for the existence of a CR function and, under certain conditions, for the realisability of the CR three-manifold. These findings are consistent with previous works by Lewandowski, Nurowski, Tafel, Hill, and independently, by Mason. We also provide an analysis of the Weyl curvature of such conformal structures in terms of the underlying CR data. In particular, we arrive at a CR formulation of the asymptotic Einstein condition by viewing conformal infinity as a cross-section of Fefferman's circle bundle.