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

TL;DR

This paper extends Bour's classical theorem to helicoidal surfaces with constant mean curvature in Bianchi-Cartan-Vranceanu spaces, characterizing their isometries and providing a classification of such surfaces.

Contribution

It generalizes Bour's theorem to BCV spaces and characterizes helicoidal surfaces with constant mean curvature within these spaces.

Findings

Existence of a two-parameter family of isometric helicoidal surfaces in BCV spaces

Characterization of constant mean curvature helicoidal surfaces in BCV spaces

Identification of minimal helicoidal surfaces in BCV spaces

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.