On the Gauss map of embedded minimal tubes
Differential Geometry
2009-03-03 v1 Geometric Topology
Abstract
A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alpha(M) between the axis and the flow vector of M. We prove that the diameter of the Gauss image of M is at least 2alpha(M). As a consequence we derive an estimate on the length of a two-dimensional minimal tube M in terms of alpha(\M) and the total Gaussian curvature of M.
Cite
@article{arxiv.0903.0228,
title = {On the Gauss map of embedded minimal tubes},
author = {Irina M. Reshetnikova and Vladimir G. Tkachev},
journal= {arXiv preprint arXiv:0903.0228},
year = {2009}
}