A characteristic property of circular cylinders

A translation surface in 3-dimensional Euclidean space is a surface that can be constructed as the sum of two regular curves alpha and beta. Recently, the minimal translation surfaces were characterized in terms of the curvature and the torsion of the generating curves. In this paper, we characterize all translation surfaces with constant and non-zero mean curvature by proving that: The only translation surface in 3-dimensional Euclidean space R-3 with constant and non-zero mean curvature H is the circular cylinder of radius 1/2 vertical bar H vertical bar.