A 1-point poly-quadrature domain of order 1 not biholomorphic to a complete circular domain

Abstract

It is known that if f : D1 → D2 is a polynomial biholomorphism with polynomial inverse and constant Jacobian then D1 is a 1-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever D2 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. arXiv:1807.08689v1 [math.CV]

Point Source Equilibrium Problems with Connections to Weighted Quadrature Domains

“…Quadrature domains whose quadrature identity involves just a multiple of a single point evaluation have been dubbed 'one-point' quadrature domains (see, for example, [5,17]). Usually the term implies that the point evaluation involves a function value, and not a derivative value.…”