Some properties of the tubular minimal surfaces of arbitrary codimension

A tubular surface is an immersion $u\colon M\to\mathbf R^n$ for which the section $\Pi\cap u(M)$ by an arbitrary hyperplane $\Pi$ orthogonal to a fixed vector $e\in\mathbf R^n$ is a compact set.
For tubular minimal surfaces in $\mathbf R^n$ we prove that
(a) if $\dim M=2$ and $u(M)$ lies in a half-space, then $u(M)$ also lies in some hyperplane; and
(b) if $\dim M\geqslant3$, then a tubular minimal surface lies in the layer between two hyperplanes orthogonal to $e$.
We obtain the corresponding results about the structure of the Gaussian image of two-dimensional tubular minimal surfaces.
The case $\operatorname{codim}M=1$ was investigated earlier (RZh.Mat., 1987, 2 B 807).
Bibliography: 19 titles.