Ribaucour transformations for constant mean curvature and linear Weingarten surfaces

Abstract

We provide a method to obtain linear Weingarten surfaces from a given such surface, by imposing a one parameter algebraic condition on a Ribaucour transformation. Our main result extends classical results for surfaces of constant Gaussian or mean curvature. By applying the theory to the cylinder, we obtain a two-parameter family of complete linear Weingarten surfaces (hyperbolic, elliptic and tubular), asymptotically close to the cylinder, which have constant mean curvature when one of the parameters vanishes. The family contains n-bubble Weingarten surfaces which are 1-periodic, have genus zero and two ends of geometric index m, where n/m is an irreducible rational number. Their total curvature vanishes, while the total absolute curvature is 8πn. We also apply the method to obtain families of complete constant mean curvature surfaces, associated to the Delaunay surfaces, which are 1-periodic for special values of the parameter.