Relative Riemann mapping criteria and hyperbolic convexity

Abstract:

Let $R$ be a simply-connected Riemann surface with a simply-connected subdomain $U$. We give a criterion in terms of conformal reflections to determine whether $R$ can be embedded in the complex plane so that $U$ is mapped onto a disc. If it can, then $U$ is convex with respect to the hyperbolic metric of $R$, by a theorem of Jørgensen. We discuss the close relationship of our criterion to two generalizations of Jørgensen’s theorem by Minda and Solynin. We generalize our criterion to the quasiconformal setting and also give a criterion for the multiply-connected case, where an embedding is sought that maps a given subdomain onto a circle domain.