@article{64aabc1a9e8d4273b95b7c678b092c20, title = "Relative Riemann mapping criteria and hyperbolic convexity", abstract = "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 J{\o}rgensen. We discuss the close relationship of our criterion to two generalizations of J{\o}rgensen'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.", keywords = "Riemann mapping, circle domains, hyperbolic convexity, conformal reflection, quasidiscs", author = "ET Crane", year = "2012", month = "7", doi = "10.1090/S0002-9939-2011-11096-7", language = "English", volume = "140", pages = "2375--2382", journal = "Proceedings of the American Mathematical Society", issn = "0002-9939", publisher = "American Mathematical Society", number = "7", }