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arXiv · 2102.00873

Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces

R. Caddeo·Irene I. Onnis·P. Piu
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Abstract

In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R^3 to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension $4$ or $6$, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

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R. Caddeo, Irene I. Onnis, P. Piu. 2021-02-01. Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces. https://arxiv.org/abs/2102.00873

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