Extensions to the boundary of Riemann maps on varying domains in the complex plane

TL;DR

The paper presents a concise proof demonstrating the boundary convergence of Riemann maps on changing domains, offering a unified method that simplifies and generalizes recent ad-hoc approaches.

Contribution

It introduces a uniform proof technique for boundary convergence of Riemann maps on varying domains, streamlining previous methods.

Findings

Provides a short, unified proof of boundary convergence.

Simplifies understanding of Riemann map behavior on varying domains.

Enhances the theoretical framework for complex analysis in variable domains.

Abstract

We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.