Some properties of the tubular minimal surfaces of arbitrary codimension

V. M. Miklyukov ; V. G. Tkachev

Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 133-150

Résumé

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.