On meromorphic functions whose image has finite spherical area

TL;DR

This paper investigates meromorphic functions with finite spherical area, establishing a limit behavior involving a tree of spheres and characterizing removable sets via extremal distance.

Contribution

It introduces a framework for understanding limits of meromorphic functions with finite spherical area and characterizes removable sets in terms of extremal distance.

Findings

Limit of meromorphic functions extends to a tree of spheres.

Removable sets are characterized by negligibility for extremal distance.

Provides a new perspective on the boundary behavior of such functions.

Abstract

In this paper, we study meromorphic functions on a domain $\Omega \subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $\Omega$ union a tree of spheres. In the second part, we show that a set $E \subset \Omega$ is removable if and only if it is negligible for extremal distance.