A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space. Introduction It is well-known that any complete nonsingular flat surface in the hyperbolic 3-space H must be a horosphere or a hyperbolic cylinder, that is, a surface equidistance from a geodesic ([20], [21]). However if we consider flat fronts (namely, projections of Legendrian immersions) and define the notion of weakly completeness, there exist many examples and interesting global properties. For instance, more recently, Martin, Umehara and Yamada [14] showed that there exists a weakly complete bounded flat front in H. The ratio ρ of canonical forms plays important roles in investigating the global properties of weakly complete flat fronts in H. Indeed, Kokubu, Rossman, Saji, Umehara and Yamada [8] showed that a point p is a singular point of a flat front in H if and only if |ρ(p)| = 1. Moreover the author and Nakajo [6] obtained the best possible upper bound for the number of exceptional values of ρ of a weakly complete flat front in H. The purpose of the present paper is to study the value distribution properties of the ratio of canonical forms of weakly complete flat fronts in H. The paper is organized as follows: In Section 1, we recall some definitions and fundamental properties of flat fronts in H, which are used throughout this paper. In Section 2, we provide a ramification theorem for the ratio of canonical forms of a weakly complete flat front in H (Theorem 2.2). The theorem is effective in the sense that it is sharp (see Corollary 3.4 and the comment below) and has some applications. We note that it corresponds to the defect relation in Nevanlinna theory ([7], [15], [16] and [18]). In Section 3, we give the two applications of this theorem. The first one is to show an analogue of a special case of the 2010 Mathematics Subject Classification. Primary 30D35, 53A35; Secondary 53C42. Date: 14 October, 2011.