# On the Gauss map of finite geometric type surfaces

## TL;DR

This paper generalizes the little Picard theorem to finite geometric type surfaces, showing their Gauss maps cannot omit more than two points, and classifies cases where the Gauss map is a regular covering.

## Contribution

It provides a topological proof of the Gauss map's limitations and classifies finite geometric type surfaces with Gauss maps as regular coverings, extending classical theorems.

## Key findings

- Gauss map cannot omit three or more points for minimal, non-flat finite geometric type surfaces.
- Classifies finite geometric type surfaces with Gauss maps as regular coverings.
- Generalizes the little Picard theorem to these surfaces.

## Abstract

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of the Euclidean three dimensional space. In a recent preprint J. Jorge and F. Mercuri gave a geometric proof that the Gauss map can not omit three or more points if the immersion is minimal and no flat. Here we give a topological proof of this result in the class of no flat finite geometric type surfaces and also give a topological classification when the Gauss map is a regular covering map. This facts are easy applications of our main result, a generalization of the little Picard theorem for the class of branched covering of a finite geometric type surface into the unit sphere of the tree dimensional Euclidean space. A finite geometric type surface given by a compact surface minus a finite set of points has the following property: any branched covering from the 0surface to the unit Euclidean sphere having a C extension to the compact surfaces can miss at most 2 points. This is a generalization of the little Picard theorem to the class of finite geometric type surfaces.