Let be a simply-connected Riemann surface with a simply-connected subdomain . We give a criterion in terms of conformal reflections to determine whether can be embedded in the complex plane so that is mapped onto a disc. If it can, then is convex with respect to the hyperbolic metric of , 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.
- Riemann mapping
- circle domains
- hyperbolic convexity
- conformal reflection
Crane, ET. (2012). Relative Riemann mapping criteria and hyperbolic convexity. Proceedings of the American Mathematical Society, 140(7), 2375-2382. [S 0002-9939(2011)11096-7]. https://doi.org/10.1090/S0002-9939-2011-11096-7