Exotic holomorphic Engel structures on C4

TL;DR

This paper constructs and classifies exotic holomorphic Engel structures on complex four-dimensional space, demonstrating their hyperbolicity and abundance of non-isomorphic examples through geometric and topological methods.

Contribution

It introduces new examples of holomorphic Engel structures on , showing their hyperbolic properties and providing a classification based on geometric and topological features.

Findings

Constructed hyperbolic Engel structures on .

Produced two infinite families of non-isomorphic structures.

Proved the existence of uncountably many non-isomorphic structures.

Abstract

A holomorphic Engel structure determines a flag of distributions $\mathcal{W}\subset \mathcal{D}\subset \mathcal{E}$. We construct examples of Engel structures on $\mathbf{C}^4$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $\mathbf{C}$. We also construct two infinite families of pairwise non-isomorphic Engel structures on $\mathbf{C}^4$ by controlling the curves $f:\mathbf{C}\to \mathbf{C}^4$ tangent to $\mathcal{W}$. The first is characterised by the topology of the set of points in $\mathbf{C}^4$ admitting $\mathcal{W}$-lines, and the second by a finer geometric property of this set. A consequence of the second construction is the existence of uncountably many non-isomorphic holomorphic Engel structures on $\mathbf{C}^4$.