Conformal Mappings for hyperbolic polygon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:51:05Z http://mathoverflow.net/feeds/question/46102 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46102/conformal-mappings-for-hyperbolic-polygon Conformal Mappings for hyperbolic polygon Marc Palm 2010-11-15T06:41:35Z 2012-10-07T14:22:47Z <p>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.</p> <p>The classical Schwarz Christoffel theorem does the job for euclidean polygons (see e.g. <a href="http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping" rel="nofollow">http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping</a>).</p> <p>Does anybody know of a similar constructions in hyperbolic geometry?</p> <p>Does anybody know of similiar constructions for any other domains?</p> <p>Any idea will be very wellcomed! I am far from being an expert in conformal mappings and do only know some isolated examples!</p> http://mathoverflow.net/questions/46102/conformal-mappings-for-hyperbolic-polygon/46269#46269 Answer by SandeepJ for Conformal Mappings for hyperbolic polygon SandeepJ 2010-11-16T18:32:06Z 2010-11-16T22:52:25Z <p>See Harmer and Martin's work on <a href="http://www.math.auckland.ac.nz/Research/Reports/view.php?id=499" rel="nofollow">Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane</a>.</p> <p>Some of the ideas developed by <a href="http://www.math.sunysb.edu/~bishop/" rel="nofollow">Christopher Bishop</a> in the context of computational geometry may also be of interest. See his <a href="http://www.math.sunysb.edu/~bishop/lectures/lec.html" rel="nofollow">talks</a> and <a href="http://www.math.sunysb.edu/~bishop/papers/papers.html" rel="nofollow">papers</a> on conformal maps.</p> http://mathoverflow.net/questions/46102/conformal-mappings-for-hyperbolic-polygon/72478#72478 Answer by Les Virany for Conformal Mappings for hyperbolic polygon Les Virany 2011-08-09T15:18:53Z 2011-08-09T15:18:53Z <p>In the Schwartz Christoffel differential vector equation, just use higher derivatives instead of first derivatives. </p> http://mathoverflow.net/questions/46102/conformal-mappings-for-hyperbolic-polygon/109065#109065 Answer by Alexandre Eremenko for Conformal Mappings for hyperbolic polygon Alexandre Eremenko 2012-10-07T14:22:47Z 2012-10-07T14:22:47Z <p>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.</p>