Uniformization by square domains

TL;DR

This paper introduces an extremal problem in conformal mapping, where the optimal map transforms a finitely connected region into a domain composed of squares aligned with axes, extending classical uniformization results.

Contribution

It formulates and solves a new extremal problem for conformal maps onto square domains, providing a novel uniformization approach for finitely connected regions.

Findings

Unique extremal conformal map onto a square domain

Characterization of the image as a union of squares with sides parallel to axes

Extension of uniformization theory to square domains

Abstract

We find an extremal problem for conformal maps on a finitely connected subregion of the Riemann sphere containing the point at infinity whose unique solution is a map onto a square domain, that is, a domain whose complementary components are (possibly degenerate) squares with sides parallel to the real or the imaginary axis.