论文信息 - The Hyperbolic Schwarz Map for the Hypergeometric Differential Equation

The Hyperbolic Schwarz Map for the Hypergeometric Differential Equation

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.

Takeshi Sasaki | Masaaki Yoshida | Kotaro Yamada

[1] Felix Klein,et al. Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade , 1884 .

[2] C. Epstein. The hyperbolic Gauss map and quasiconformal reflections. , 1986 .

[3] Takeshi Sasaki,et al. Confluence of Swallowtail Singularities of the Hyperbolic Schwarz Map Defined by the Hypergeometric Differential Equation , 2008, Exp. Math..

[4] Masaaki Yoshida,et al. Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts , 2007 .

[5] A. Martínez,et al. Flat surfaces in the hyperbolic 3-space , 2000 .

[6] Kentaro Saji,et al. Singularities of flat fronts in hyperbolic space , 2004 .

[7] Flat fronts in hyperbolic 3-space , 2003, math/0301224.

[8] 岩崎克則,et al. From Gauss to Painlevé : a modern theory of special functions : dedicated to Tosihusa Kimura , 1991 .

[9] R. M. EL-ASHWAH,et al. HYPERGEOMETRIC FUNCTIONS , 2004 .

[10] J. M. Boardman,et al. Singularties of differentiable maps , 1967 .

[11] Masaaki Umehara,et al. An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C) , 2002 .