The Hyperbolic Schwarz Map for the Hypergeometric Differential Equation

Abstract

The Schwarz map of the hypergeometric differential equation has been 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 hyperbolic 3-space. This map can be considered to be a lifting to 3-space of the Schwarz map. In this paper, we study the singularities of this map, and attempt to visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in forthcoming papers.

Keywords:

• 33C05, 53C42
• hypergeometric differential equation
• Schwarz map
• hyperbolic Schwarz map
• flat surfaces
• flat fronts