Experimental Mathematics

The Hyperbolic Schwarz Map for the Hypergeometric

Takeshi Sasaki, Kotaro Yamada, and Masaaki Yoshida

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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.

Article information

Experiment. Math., Volume 17, Issue 3 (2008), 269-282.

First available in Project Euclid: 19 November 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33C05: Classical hypergeometric functions, $_2F_1$ 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

hypergeometric differential equation Schwarz map hyperbolic Schwarz map flat surfaces flat fronts


Sasaki, Takeshi; Yamada, Kotaro; Yoshida, Masaaki. The Hyperbolic Schwarz Map for the Hypergeometric. Experiment. Math. 17 (2008), no. 3, 269--282. https://projecteuclid.org/euclid.em/1227121382

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