Hyperbolic contact symplectic lifts
Consider a holomorphic contact manifold. Holomorphic discs tangent to the contact planes define a pseudometric on the manifold. This pseudometric integrates to a pseudodistance. When the pseudodistance is a distance, we call the contact manifold \emph{contact-hyperbolic}, by analogy with Kobayashi hyperbolicity. The goal of this paper is to construct explicit examples of contact-hyperbolic contact manifolds with large automorphism groups. We study Reeb manifolds: holomorphic contact structures equipped with a Reeb vector field whose flow acts freely. Our first main theorem shows that every proper Reeb manifold admits a holomorphic symplectic quotient. It also identifies which symplectic manifolds arise this way. The isomorphism classes of proper Reeb manifolds over a fixed symplectic base manifold are parameterised by the first cohomology. Our second main theorem: a proper Reeb manifold is (complete) contact-hyperbolic if and only if its symplectic quotient manifold is (complete) Kobayashi hyperbolic. This theorem allows us to construct many new explicit examples of contact-hyperbolic contact manifolds. Finally, we study the group of contact biholomorphisms. Contact hyperbolicity implies that this group is a finite-dimensional Lie group. For contact $3$-manifolds, we sharply bound the dimension of the automorphism group. We give examples with automorphism groups reaching every possible dimension. Our third main theorem: the unique maximally symmetric example, up to isomorphism, is the contact manifold $\mathbb B^2_{z,w}\times\mathbb C_y$ with contact form $dy+(1-z)^{-2}dw$.