Worm domains and Fefferman space-time singularities

Let $\mathcal{W}$ be a smoothly bounded worm domain in $\mathbb{C}^2$ and let $\mathcal{A} = Null(L_\theta)$ be the set of Levi-flat points on the boundary $\partial \mathcal{W}$ of $\mathcal{W}$. We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus $M = \partial \mathcal{W} \setminus \mathcasl{A}$ and the theory of space–time singularities associated to the Fefferman metric $F_\theta$ on the total space of the canonical circle bundle $S^1 \to C(M) \stackrel{\pi}{\to} M$. Given any point $(0,w_0) \in \mathcal{A}$, we show that every lift $\Gamma (\varphi) \in C(M)$, $0 \leq \varphi − \log |w_0|^2 < \pi/2$, of the circle $\Gamma_{w_0} : r = 2 \cos [\log |w_0|^2 − \varphi]$ in $M$, runs into a curvature singularity of Fefferman’s space–time $(C(M), F_\theta )$. We show that $\Sigma = \pi^{-1}(\Gamma_{w_0})$ is a Lorentzian real surface in $(C(M), F_\theta )$ such that the immersion $\iota : \Sigma \hookrightarrow C(M)$ 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 $O(1, 1) \to O(\Sigma ) \to \Sigma$ and adapted Lorentzian frames $O(1, 1) \times O(2) \to O(C(M),\Sigma) \to \Sigma$, 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(\buildrel{\cdot}\over{\Sigma} ) \subset \partial_{\mathrm{adt}} C(M)$.