Abstract
In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space {mathbb {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.
Highlights
Introduction and preliminariesHelicoidal surfaces in the Euclidean three-dimensional space R3 are invariant under the action of the 1-parameter group of helicoidal motions and are a generalization of rotation surfaces
Since the beginning of differential geometry of surfaces, much attention has been given to the surfaces of revolution with constant Gauss curvature or constant mean curvature (CMC-surfaces)
Helicoidal minimal surfaces were studied by Scherk in 1835, but it is rather recent the classification of the helicoidal surfaces in R3 with nonzero constant mean curvature, given by Do Carmo and Dajczer in [18]
Summary
Helicoidal surfaces in the Euclidean three-dimensional space R3 are invariant under the action of the 1-parameter group of helicoidal motions and are a generalization of rotation surfaces. By using the result of Bour, in [18] Do Carmo and Dajczer established a condition for a surface of the Bour’s family to have constant mean curvature They obtained an integral representation (depending on three parameters) of helicoidal surfaces with nonzero constant mean curvature, which is a natural generalization of the representation for Delaunay surfaces, i.e., CMC rotation surfaces, given by Kenmotsu (see [21]). 4, we use techniques of equivariant geometry, in particular the Reduction Theorem of Back, do Carmo and Hsiang (see [2]), to deduce a differential equation that the function U(u) must satisfy in order that a helicoidal surface of the Bour’s family determined by U(u) has constant mean curvature We solve this equation by making a transformation of coordinates, treating separately the case of the space forms R3 and 3 from the other BCV spaces. We conclude by showing that in R3 these results give a natural parametrization of all the helicoidal minimal surfaces obtained by Scherk in [37]
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