Open Access
Translator Disclaimer
April, 2009 Hyperbolic Schwarz map of the confluent hypergeometric differential equation
Kentaro SAJI, Takeshi SASAKI, Masaaki YOSHIDA
J. Math. Soc. Japan 61(2): 559-578 (April, 2009). DOI: 10.2969/jmsj/06120559


The hyperbolic Schwarz map is defined in [SYY1] as a map from the complex projective line to the three-dimensional real hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front ([GMM], [KUY], [KRSUY]), and generic singularities are cuspidal edges and swallowtail singularities. In this paper, for the two-parameter family of the confluent hypergeometric differential equations, we study the singularities of the hyperbolic Schwarz map, count the number of swallowtails, and identify the further singularities, except those which are apparently of type A5. This describes creations/eliminations of the swallowtails on the image surfaces, and gives a stratification of the parameter space according to types of singularities. Such a study was made for a 1-parameter family of hypergeometric differential equation in [NSYY], which counts only the number of swallowtails without identifying further singularities.


Download Citation

Kentaro SAJI. Takeshi SASAKI. Masaaki YOSHIDA. "Hyperbolic Schwarz map of the confluent hypergeometric differential equation." J. Math. Soc. Japan 61 (2) 559 - 578, April, 2009.


Published: April, 2009
First available in Project Euclid: 13 May 2009

zbMATH: 1205.33007
MathSciNet: MR2532901
Digital Object Identifier: 10.2969/jmsj/06120559

Primary: 33C05 , 53C42

Keywords: confluent hypergeometric differential equation , flat front , hyperbolic Schwarz map , swallowtail singularity

Rights: Copyright © 2009 Mathematical Society of Japan


Vol.61 • No. 2 • April, 2009
Back to Top