Worm domains and Fefferman space-€“time singularities

Let W be a smoothly bounded worm domain in C2 and let A=Null(Lθ) be the set of Levi-flat points on the boundary ∂W of W. We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus M=∂W∖A and the theory of space–time singularities associated to the Fefferman metric Fθ on the total space of the canonical circle bundle S1→C(M)⟶πM. Given any point (0,w0)∈A, we show that every lift Γ(φ)∈C(M), 0≤φ−log|w0|2<π∕2, of the circle Γw0: r=2cos[log|w0|2−φ] in M, runs into a curvature singularity of Fefferman's space–time (C(M),Fθ). We show that Σ=π−1(Γw0) is a Lorentzian real surface in (C(M),Fθ) such that the immersion ι:Σ↪C(M) has a flat normal connection. Consequently, there is a natural isometric immersion j:O(Σ)→O(C(M),Σ) between the total spaces of the principal bundles of Lorentzian frames O(1,1)→O(Σ)→Σ and adapted Lorentzian frames O(1,1)×O(2)→O(C(M),Σ)→Σ, endowed with Schmidt metrics, descending to a map of bundle completions which maps the b-boundary of Σ into the adapted bundle boundary of C(M), i.e. j(Σ̇)⊂∂adtC(M).