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

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.

22 pages. Result about the missing points by the Gauss map of a wide class of complete parabolic surfaces in the 3 dimensional Euclidean space with finite total curvature and the Gauss map extending continuously to a compact Riemann surfaces including the minimal surfaces. If it is no flat then the Gauss map can not miss more than 2 points of the sphere