Abstract

In 1976, Charles Fefferman constructed, in a canonical way, a Lorentzian conformal structure on a circle bundle over a given strictly pseudoconvex Cauchy-Riemann (CR) manifolds of hypersurface type. It is also known, notably through the work of Sir Roger Penrose and his associates, and that of the Warsaw group led by Andrzej Trautman, that CR three-manifolds underlie Einstein Lorentzian four-manifolds whose Weyl tensors are said to be algebraically special. I will show how these two perspectives are related to each other, by presenting modifications of Fefferman’s original construction, where the conformal structure is “perturbed” by some semi-basic one-form, which encodes additional data on the CR three-manifold. Metrics in such a perturbed Fefferman conformal class whose Ricci tensor satisfies certain degeneracy conditions are only defined off sections of the Fefferman bundle, which may be viewed as “conformal infinity”. The prescriptions on the Ricci tensor can then be reduced to differential constraints on the CR three-manifold in terms of 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, closely related to so-called BGG operators. A solution thereof provides a criterion for the existence of a CR function and, under certain conditions, for CR embeddability. Our setup allows us to reinterpret previous works by Lewandowski, Nurowski, Tafel, Hill, and independently, by Mason. Time permitting, I will discuss the higher-dimensional story. This talk is based on arXiv:2303.07328 and arXiv:2309.16986.