On the image of the Gauss map of an immersed surface with constant mean curvature in R-3

We prove, generalizing a veil known property of Delaunay surfaces, that if the Gauss image of a cmc surface in the Euclidean space is a compact surfac e with boundary, then any connected component of sphere minus the image is a strictly convex domain. We also obtain conditions under which the Gauss i mage has a regular boundary. These results relate to the question, raised b y do Carmo, of whether the Gauss image of a complete cmc surface contains a n equator of the sphere.