Remarks on the Darboux Transform of Isothermic Surfaces

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.

1991 Mathematics Subject Classification: (Primary) 53A10, (Secondary) 53A50, 53C42.

Keywords: Isothermic surface, Darboux transformation, Christoffel transformation, Riccati equation, Constant mean curvature, Baecklund transformation.

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