Given a complex manifold M endowed with a holomorphic contact structure V, one can define a sub-Finsler pseudometric through holomorphic discs tangent to V . If the integrated pseudodistance is a distance, then M is said to be V -hyperbolic, a generalisation of the classical notion of Kobayashi hyperbolicity. Whenever M is Kobayashi hyperbolic, it is also V-hyperbolic with respect to any contact structure V. However, there are very few  non-trivial examples of hyperbolic contact structure (a remarkable one was constructed on C^3 by Franc Forstnerič).

In this talk, I will focus on Reeb manifolds M, which are contact manifolds with a free holomorphic C-action generated by a Reeb vector field. Every proper Reeb manifold is the total space of a contact symplectic lift, that is, it admits a C-principal bundle structure onto a complex manifold S endowed with a C∞-exact holomorphic symplectic form ω so that the pull back of ω is related to the contact structure V of M. Conversely,  any C∞-exact holomorphic symplectic manifold is given as the base of a contact symplectic lift. It follows that M is (complete) V -hyperbolic if and only if S is (complete) Kobayashi hyperbolic. This yields many new (non-trivial) examples of V-hyperbolic manifolds.  For instance, if S is a pseudoconvex bounded domain in C^{2N}, then SxC is V-hyperbolic with respect to the standard contact structure given by dy+\sum z_j dw_j (here (z,w) are coordinates on S and y on C). If S is also strongly pseudoconvex than SxC is complete V-hyperbolic. Also, the only (up to natural equivalence) V-hyperbolic proper Reeb manifold of dimension 3 with V-preserving automorphism (real Lie) group of maximal dimension (i.e. 8) is the lift  of the symplectic form dz \wedge dw/(1-z)^3 on the unit ball of C^2.

The talk is based on a joint work with B. McKay and R. Ugolini.