Hyperbolic Schwarz map for the hypergeometric differential equation

Takeshi Sasaki, Kotaro Yamada, Masaaki Yoshida

Published 2006-09-07, updated 2007-02-28Version 2

The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in a forthcoming paper.