Confluence of Swallowtail Singularities of the Hyperbolic Schwarz Map Defined by the Hypergeometric Differential Equation

The papers [Gálvez et al. 2000, Kokubu et al. 2003, Kokubu et al. 2005] gave a method of constructing flat surfaces in the three-dimesnional hyperbolic space. Such surfaces have generically singularities, since any closed nonsigular flat surface is isometric to a horosphere or a hyperbolic cylinder. In the paper [Sasaki et al. 2006], we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front and its generic singularities are cuspidal edges and swallowtail singularities. In this paper we study the curves consisting of cuspidal edges and creation/elimination of swallowtail singularities depending on the parameters of the hypergeometric equation.