On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto's theorem (Q1116462)

The main results of this paper are: Theorem 1. Let S be a complete minimal surface in \({\mathbb{R}}^ 3\). If the Gauss map g takes on 5 distinct values only a finite number of times, then S has finite total curvature. Theorem 2. Let S be a complete minimal surface in \({\mathbb{R}}^ 3\), not a plane. If the Gauss map g of S omits 4 points on the unit sphere, then every other point must be covered infinitely often. Theorem 3. Let S be a complete minimal surface in \({\mathbb{R}}^ 4\), not a plane. If each of the factors \(g_ k\) of the Gauss map of S omits 4 distinct points, then each of the \(g_ k\) must cover every other point infinitely often. If one of the \(g_ k\) is constant, then the other must cover every point infinitely often with at most 3 exceptions. - Examples are given to show that all of these statements are sharp.