The Gauss map of a complete minimal surface with finite total curvature

TL;DR

This paper improves the understanding of the Gauss map of complete minimal surfaces with finite total curvature, showing it omits at most two points, refining previous bounds and extending to a broader class of immersions.

Contribution

It proves that the Gauss map of such surfaces omits at most two points, sharpening Osserman's previous bound of three and applying to a wider class of isometric immersions.

Findings

Gauss map omits at most two points for these surfaces

The result is sharp, as exemplified by the catenoid

Extension of the result to a broader class of immersions

Abstract

In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits exactly two points. In fact we prove this result for a wider class of isometric immersions, that share the basic differential topological properties of the complete minimal surfaces of finite total curvature.