I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job for euclidean polygons (see e.g. http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping).
Does anybody know of a similar construction constructions in hyperbolic geometry?
Does anybody know of similiar construction constructions for any other domains?
Any idea , how to construct conformal mappings even for more general domains will also be very wellcomed! Perhaps I can spezialise them to my preferred domains. I am far from being an expert in conformal mappings and do only know some isolated examples!