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

Abstract

The papers [Gálvez et al. 00, Kokubu et al. 03, Kokubu et al. 05] gave a method of constructing flat surfaces in threedimensional hyperbolic space. Generically, such surfaces have singularities, since any closed nonsingular flat surface is isometric to a horosphere or a hyperbolic cylinder. In [Sasaki et al. 08a], we defined a map, called the hyperbolic Schwarz map, from one-dimensional projective space to three-dimensional hyperbolic space using solutions of the Gauss 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 the creation and elimination of swallowtail singularities depending on the parameters of the hypergeometric equation.