Worm domains and Fefferman space–time singularities

Abstract

Let W be a smoothly bounded worm domain in C 2 and let A = Null ( L θ ) 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 F θ on the total space of the canonical circle bundle S 1 → C ( M ) ⟶ π M . Given any point ( 0 , w 0 ) ∈ A , we show that every lift Γ ( φ ) ∈ C ( M ) , 0 ≤ φ − log | w 0 | 2 < π ∕ 2 , of the circle Γ w 0 : r = 2 cos [ log | w 0 | 2 − φ ] in M , runs into a curvature singularity of Fefferman’s space–time ( C ( M ) , F θ ) . We show that Σ = π − 1 ( Γ w 0 ) is a Lorentzian real surface in ( C ( M ) , F θ ) 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 ( Σ ̇ ) ⊂ ∂ adt C ( M ) .