The distribution of normal vectors of minimal surfaces

Question

Is there a minimal surface such that its normal vectors are distributed everywhere on the unite sphere? I've got the results above:

1. Let $S$ be a complete regular minimal surface in $\mathbb{E}^3$. Then either $S$ is a plane or else the normal vectors to $S$ are everywhere dense.
2. Let $S$ be a complete regular minimal surface in $\mathbb{E}^3$. Then either $S$ is a plane, or else the set $E$ omitted by the image of $S$ under the Gauss map has capacity zero.
3. Let $E$ be an arbitrary set of $k$ points on the unit sphere, where $k\le 4$. Then there exists a complete regular minimal surface in $\mathbb{E}^3$ whose image under the Gauss map omits precisely the set $E$.