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

An improved Riemann Mapping Theorem and complexity in potential theory