An improved Riemann Mapping Theorem and complexity in potential theory

TL;DR

This paper presents an enhanced version of the Riemann mapping theorem that applies to finitely connected domains, using double quadrature domains to better understand potential theory complexities.

Contribution

It introduces an improved Riemann mapping theorem for finitely connected domains using double quadrature domains, expanding the classic theorem's applicability and providing new insights into potential theory.

Findings

The biholomorphic map can be made close to the identity for domains with smooth boundaries.

Double quadrature domains approximate original domains closely in the improved theorem.

The results illuminate the complexity of potential theory objects in multiply connected domains.

Abstract

We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another "double quadrature domain," i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann Mapping Theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains.