Holomorphic Hulls for Compact 3-Manifolds

We demonstrate the different possible structures for holomorphic hulls for embeddings of compact real 3-manifolds $M\hookrightarrow {\mathbb{C}}^3$ along the set of complex tangents $\gamma$. Using our previous work [1], we can construct embeddings with any prescribed link and 2-plane field along it, as well as a prescription of angles at each point that determines the Bishop invariant. We show that elliptic points ($\gamma <\frac{1}{2}$) produce analytic discs filling a Levi-flat hypersurface, hyperbolic points ($\gamma >\frac{1}{2}$) add no local hull structure, and parabolic points ($\gamma =\frac{1}{2}$) may develop a “delicate” Jöricke `onion’ structure. We illustrate these phenomena with explicit examples of parabolic knots and links with mixed components.