On complete minimal surfaces whose Gauss map misses two directions (Q1345840)

Many examples of complete minimal surfaces with finite total curvature, whose Gauss maps miss two directions, are constructed. Among them there exist two series of generalized Costa surface displayed using computer graphics (Theorem 3). They are of the conformal type of a torus minus 4 points, while a Costa surface, a special case of series, has the conformal type of a torus minus 3 points and one missing normal direction. As a conclusion, there exist complete minimal surfaces with finite total curvature, whose Gauss maps miss two directions, for surfaces of the conformal tye of \(\widehat{M}_ k - \{r \text{ points}\}\), where \(\widehat{M}_ k\) is a compact surface of genus \(k\), if (1) \(r \geq 2\) when \(k = 0\), (2) \(r \geq 3\) when \(k = 1\), (3) \(r \geq 4\) when \(k \geq 2\) (Theorem 4). This is an approach to the problem of getting sharp estimates of the number of missing normal directions of minimal surfaces with finite total curvature.