A Riemann surface attached to domains in the plane and complexity in potential theory (Q2725481)

Let \(\Omega\) be a finitely connected domain in the complex plane such that no boundary component is a point.NEWLINENEWLINENEWLINEIt was proved by the author [S. Bell, Duke Math. J. 98, No. 1, 187-207 (1999; Zbl 0948.30015)] that the Bergman kernel \(K(z,w)\) of \(\Omega\) is algebraic if and only if the Szegő kernel is algebraic.NEWLINENEWLINENEWLINEIn the present paper, it is shown that algebraicity of either of them implies existence of a compact Riemann surface \(R\) such that \(\Omega\) is a domain in \(R\) and the kernels \(K(z,w)\) and \(S(z,w)\) (together with other classical domain functions) extend to \(R\) as single valued meromorphic functions. As a result, there exist two holomorphic functions \(f\) and \(g\) in \(\Omega\) such that all these domain functions and all proper holomorphic mappings of \(\Omega\) onto the unit disk are rational combinations of \(f(z),g(z), \overline{f(w)}, \overline{g(w)}\). Some open questions are discussed.