On the Gauss map of embedded minimal tubes

zbMATH Open1023.53047arXiv0903.0228MaRDI QIDQ4799204

I. M. Reshetnikova, V. G. Tkachev

Publication date: 18 March 2003

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.

Full work available at URL: https://arxiv.org/abs/0903.0228

zbMATH Keywords

Gaussian maptotal Gaussian curvatureflow vectorminimal tubes

Mathematics Subject Classification ID

Higher-dimensional and -codimensional surfaces in Euclidean and related (n)-spaces (53A07) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42)

Related Items (1)

Minimal tubes of finite integral curvature

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