Remarks on the Darboux transform of isothermic surfaces

Abstract

We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean space. Using quaternionic calculus we derive a Riccati type equation which characterizes all Darboux transforms of a given isothermic surface. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: their parallel surfaces of constant mean curvature are Christoffel and Darboux transforms at the same time. We prove – as a generalization of Bianchi's theorem on minimal Darboux transforms of minimal surfaces – that constant mean curvature surfaces in Euclidean space allow ${\infty }^3$ Darboux transforms into surfaces of constant mean curvature. We indicate the relation between these Darboux transforms and Bäcklund transforms of spherical surfaces.