A property of complete minimal surfaces

Abstract:

If $M$ is a complete minimal surface in ${R^n}$, we denote by $W$ the set of points in ${R^n}$ that do not lie on any tangent plane of $M$. By taking a point in $W$ as origin, the position vector of $M$ determines a global unit normal vector field $e$ to $M$. We prove that if $e$ is a minimal section, then $M$ is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in ${R^3}$ covers all ${R^3}$. We also prove a similar result for a complete minimal surface $M$ in ${S^3}$, and deduce from it that if the spherical image of $M$ lies in a closed hemisphere, then $M$ is a great ${S^2}$.