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

TL;DR

This paper establishes a ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space, leading to applications including an Ahlfors islands theorem analogue and a classification of complete nonsingular flat surfaces.

Contribution

It introduces an effective ramification theorem for the ratio of canonical forms, providing new tools for analyzing flat surfaces in hyperbolic space.

Findings

Proves an analogue of the Ahlfors islands theorem for flat fronts in hyperbolic space.

Provides a simple proof of the classification of complete nonsingular flat surfaces.

Establishes a ramification theorem that enhances understanding of the geometric structure.

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.