Unbounded visibility domains: metric estimates and an application

TL;DR

This paper establishes explicit lower bounds for the Kobayashi metric on certain smooth pseudoconvex domains, constructs unbounded hyperbolic domains with negative curvature properties, and proves a Picard-type extension theorem.

Contribution

It provides a new lower bound estimate for the Kobayashi metric using complex Monge-Ampère equation regularity, and constructs unbounded hyperbolic domains with curvature properties.

Findings

Derived explicit Kobayashi metric lower bounds in smooth pseudoconvex domains.

Constructed unbounded Kobayashi hyperbolic domains with negative curvature.

Proved a Picard-type extension theorem for these domains.

Abstract

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.