Engel CR submanifolds of $\mathbb{C}^3$

Abstract: We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg.