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We describe the space Σ H of all surfaces in R z that have constant mean curvature HφO and are invariant by helicoidal motions, with a fixed axis, of R Ά . Similar to the case Σ o of minimal surfaces Σ H behaves roughly like a circular cylinder where a certain generator corresponds to the rotation surfaces and each parallel corresponds to a periodic family of isometric helicoidal surfaces.1. Introduction. 1.1. Rotation surfaces in the Euclidean space ϋ! 3 with constant mean curvature have been known for a long time (Delaunay [3]). A natural generalization of rotation surfaces are the helicoidal surfaces that can be defined as follows.Let R 3 have coordinates (x, y, z). Consider the one-parameter subgroup g t :R3 -> R 3 of the group of rigid motions of R 3 given by 9t(%> V, z) = (& cos t + y sin t, -x sin t + y cos t, z + ht) , te(-oo f oo).The motion g t is called a helicoidal motion with axis Oz and pitch h.A helicoidal surface with axix Oz and pitch pitch h is a surface that is invariant by g t , for all t. When h = 0, they reduce to rotation surfaces.The helicoidal minimal surfaces have also been known for quite a long time (see e.g. [6] for details). It is therefore mildly surprising that we do not find in the literature the helicoidal surfaces with constant nonzero mean curvature; in this paper we want to determine explicitly all of them.Our interest in this question comes (aside its naturality) from the fact that there are very few explicit examples of surfaces with nonzero constant mean curvature. To understand certain aspects of such surfaces (behaviour of the Gauss map, stability, etc.) it might prove convenient to have at hand a reasonable supply of explicit examples. It should be mentioned that the techniques used here can also give a complete description of helicoidal surfaces with constant Gaussian curvature (Remark 3.16); this is, however, very simple and probably known.

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