Unbounded visibility domains: metric estimates and an application

We give an explicit lower bound, in terms of the distance from the boundary, for the Kobayashi metric of a certain class of bounded pseudoconvex domains in ${\mathbb{C}}^n$ with ${\mathcal{C}}^2$-smooth boundary using the regularity theory for the complex Monge–Ampere equation. Using such an estimate, among other tools, we construct a family of unbounded Kobayashi hyperbolic domains in ${\mathbb{C}}^n$ having a certain negative-curvature-type property with respect to the Kobayashi distance. As an application, we prove a Picard-type extension theorem for the latter domains.