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

Abstract

Abstract. 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.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

“…We here call it an improper affine front because Nakajo [32] and Umehara and Yamada [40] showed that an improper affine map is a front in R 3 . Moreover, we [22] gave similar result for flat fronts in H 3 . In [21], we obtained a geometric interpretation for the maximal number of exceptional values of their Gauss maps.…”

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

“…As an application of this result, a simple proof of the parametric affine Bernstein theorem for improper affine spheres in R 3 was provided. Moreover, the authors [17,19] proved similar results for flat fronts in the hyperbolic three-space H 3 .…”

“…As a corollary of Theorem 3.8, we give a simple proof of the classification ( [35], [39]) of complete nonsingular flat surfaces in H 3 . For the proof, see [17,Corollary 3.5].…”

Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded

“…Also, the assumption of weak completeness in the theorem is crucial, since two hyperbolic Gauss maps can omit at most finite points if the given flat front is complete (see Remark 2.5). In contrast to this theorem, Kawakami [7] showed the ratio of canonical forms ρ of weakly complete flat fronts can omit at most three exceptional values.…”