Worm domains and Fefferman space-time singularities

Let W be a smoothly bounded worm domain in C-2 and let A = Null(L-theta) be the set of Levi-flat points on the boundary partial derivative W of W. We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus M = partial derivative W \ A and the theory of space-time singularities associated to the Fefferman metric F-theta on the total space of the canonical circle bundle S-1 -> C(M) -> (pi) M. Given any point (0, w(o)) is an element of A, we show that every lift Gamma(phi) is an element of C(M), 0 <= phi - log |w(o)|(2) < pi/2, of the circle Gamma(w0) : r = 2 cos[log |w(o)|(2) - phi] in M, runs into a curvature singularity of Fefferman's space-time (C(M), F-theta). We show that Sigma = pi(-1) (Gamma(w0)) is a Lorentzian real surface in (C(M), F-theta) such that the immersion l : Sigma curved right arrow C(M) has a flat normal connection. Consequently, there is a natural isometric immersion j : O(Sigma) -> O(C(M), Sigma) between the total spaces of the principal bundles of Lorentzian frames O(1, 1) -> O(Sigma) -> Sigma and adapted Lorentzian frames O(1, 1) x O(2) -> O(C(M), Sigma) -> Sigma, endowed with Schmidt metrics, descending to a map of bundle completions which maps the b-boundary of Sigma into the adapted bundle boundary of C(M), i.e. j(Sigma) over dot subset of partial derivative(adt) C(M). (C) 2017 Elsevier B.V. All rights reserved.