Visibility property in one and several variables and its applications

TL;DR

This paper investigates the visibility property related to the Kobayashi distance in complex domains, establishing new results on boundary removability, domain classifications, and extensions of conformal and biholomorphic maps, with implications for the MLC conjecture.

Contribution

It provides new characterizations of visibility in complex domains, including localization results, boundary extension conditions, and a reformulation of the MLC conjecture in terms of visibility.

Findings

Totally disconnected boundary sets are removable for visibility.

A domain is a local weak visibility domain iff it is a weak visibility domain.

Visibility implies compactness of the end topology.

Abstract

In this paper we report our investigations on visibility with respect to the Kobayashi distance and its applications, with a special focus on planar domains. We prove that totally disconnected subsets of the boundary are removable in the context of visibility. We also show that a domain in $\mathbb{C}^n$ is a local weak visibility domain if and only if it is a weak visibility domain. The above holds also for visibility. Along the way, we prove an intrinsic localization result for the Kobayashi distance. Moreover, we observe some interesting consequences of weak visibility; for example, weak visibility implies compactness of the end topology of the closure of the domain. For planar domains: (i) We provide examples of visibility domains that are not locally Goldilocks at any boundary point. (ii) We provide certain general conditions on planar domains that yield the continuous extension of conformal maps, generalizing the Carath\'{e}odory extension theorem. Our conditions are quite general and assume very little regularity of the boundary. We demonstrate this through examples. (iii) We also provide conditions for the homeomorphic extension of biholomorphic maps up to the boundary. (iv) We prove that a hyperbolic, simply connected domain possesses the visibility property if and only if its boundary is locally connected. This leads us to reformulate the MLC conjecture in terms of visibility. (v) We provide a characterization of visibility for a large class of planar domains including certain uncountably connected domains.