A domain \(\Omega\subset \mathbb{C}\) is a quadrature domain (in the classic sense) if \[ \int_\Omega f dA=\sum_{k=1}^m\sum_{j=0}^{m_k} c_{mj}f^{(j)}(a_k) \] holds for all analytic functions in \(\Omega\), and where the nodes \(\{a_k\}\) belong to \(\Omega\). It is known that such type of quadrature domains have an algebraic boundary. So by Green’s theorem the above quadrature identity is equivalent to \[ \int_\Omega f dA=\frac{1}{2}\oint_{\partial\Omega}f(z)r_\Omega(z)dz, \] where \[ r_\Omega(z)=\pi^{-1}\sum_{k=1}^m\sum_{j=0}^{m_k} c_{mj}\frac{j !}{(z-a_k)^{j+1}}. \]
The main achievement of the paper is a bound on the connectivity of \(\Omega\) in terms of the degree of the rational function \(r_\Omega\) and the number of poles of that functions. Previous bounds were obtained by B. Gustafsson [J. Anal. Math. 51, 91–117 (1988: Zbl 0656.30034)], and the present paper improves his results significantly. The authors consider both bounded and unbounded quadrature domains, but under the assumptions that the boundary is compact. This assumption allows to apply acircular inversion in order to transfer unbounded quadrature domains into bounded and vice versa.
It is well known that the boundary of a quadrature domain admits a Schwarz function, that is, a holomorphic function in a neighborhood of the boundary that equals to \(\bar z\) on the boundary and has poles at the nodes \(\{a_k\}\). The investigation of this function on the Schottky double of \(\Omega\) is an essential ingredient of the proof. The Schottky double of \(\Omega\) is a compact Riemann surface with conformal structure on \(\Omega\) and an anti-conformal structure on the “back” of a copy of \(\Omega\). The implementation of this technique to quadrature domains was introduced by B. Gustafsson [Acta Appl. Math. 1, 209–240 (1983: Zbl 0559.30039)]. In addition to that tool, the authors use a method from complex dynamics to determinate the number of zeros of certain harmonic polynomials, and a perturbation technique which is based on the Hele-Shaw flow.