# Uniform convergence of Green's functions

## TL;DR

This paper proves that Green's functions for a sequence of regular planar domains converge uniformly on the complex sphere under kernel convergence, with conditions on regularity and bounded connectivity.

## Contribution

It establishes uniform convergence of Green's functions for converging planar domains, extending previous results to more general settings.

## Key findings

- Green's functions converge uniformly on the complex sphere
- Convergence holds under kernel convergence with regularity and bounded connectivity
- Results apply to sequences of regular planar domains

## Abstract

Given a sequence of regular planar domains converging in the sense of kernel, we prove that the corresponding Green's functions converge uniformly on the complex sphere, provided the limit domain is also regular, and the connectivity is uniformly bounded.