On the Gauss map of embedded minimal tubes

TL;DR

This paper investigates the relationship between the geometry of the Gaussian image of higher-dimensional minimal tubes and their flow vector, providing bounds on the Gauss image diameter and estimates on the tube length based on geometric parameters.

Contribution

It introduces a new geometric relationship linking the Gauss image diameter of minimal tubes to the angle between the axis and flow vector, with implications for tube length estimates.

Findings

Diameter of Gauss image ≥ 2 * alpha(M)

Length of minimal tube estimated by alpha(M) and total Gaussian curvature

Flow vector invariance in zero mean curvature tubes

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.