Bianchi-Backlund transformations for constant mean curvature surfaces with umbilics theory and applications

The Bianchi-Backlund transformations are an extension to surfaces of positive Gauss curvature and constant mean curvature of the Backlund transformations for surfaces of negative curvature. The classical construction depends on the existence of curvature lines at all points of the surface. Our goal is to extend theses transformations to surfaces with umbilic points. We show that using a coordinate transformation away from the umbilics, we obtain a new system of equations whose solutions extend to the umbilics. Additionally, associated to the Backlund (and Bianchi-Backlund) transformations is the concept of a dressing action. We show that away from umbilics, the classical Bianchi-Backlund transformations are a special case of the dressing action used in the theory of harmonic maps. We use as an application a class of surfaces of constant mean curvature possessing an intrinsic rotational symmetry and umbilics and we apply the Bianchi-Backlund transformations to obtain new surfaces.