Conformal Mappings for hyperbolic polygon 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 constructions in hyperbolic geometry?
Does anybody know of similiar constructions for any other domains?
Any idea will be very wellcomed! I am far from being an expert in conformal mappings and do only know some isolated examples!
 A: See Harmer and Martin's work on Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane.
Some of the ideas developed by Christopher Bishop in the context of computational geometry may also be of interest. 
See his talks and papers on conformal maps.
A: There is a theory of conformal map for circular polygons (polygons bounded by arcs of circles).
But in this case, instead of an integral in the Schwarz-Christoffel formula, you obtain a linear
differential equation. In the case of a circular triangle, the equation is hypergeometric and you
have an explicit representation of your mapping. The paper of Harmer and Martin mentioned in the previous answer deals mainly with the case of a triangle. The most comprehensive treatment of triangles
is in the second volume of Caratheodory's textbook on complex variables, and in other books
on hypergeometric functions. The case of quadrilateral is
the simplest case when there is no explicit formula. It was subject of much research.
See, for example, arXiv:1110.2696,  arXiv:1111.2296, and references in these papers.
A: In the Schwartz Christoffel differential vector equation, just use higher derivatives instead of first derivatives. 
