A $1$-point Quadrature domain of order $1$ not biholomorphic to a balanced domain

Abstract

It is known that if $f: D_1 \to D_2$ is a polynomial biholomorphism with polynomial inverse and constant Jacobian then $D_1$ is a $1$-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order $1$ whenever $D_2$ is a balanced domain. Bell conjectured that all $1$-point Quadrature domains arise in this manner. In this note, we construct a $1$-point Quadrature domain of order $1$ that is not biholomorphic to any balanced domain.