THE μ–DARBOUX TRANSFORMATION OF MINIMAL SURFACES

The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated C∗–family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so–called µ–Darboux transforms. We show that a µ–Darboux transform of a minimal surface is not minimal but a Willmore surface in 4–space. More precisely, we show that a µ–Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in CP3 which is canonically associated to a minimal surface fp,q in the right–associated family of f . Here we use an extension of the notion of the associated family fp,q of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at µ = 1. Moreover, the family of Willmore surfaces µ–Darboux transforms, µ ∈ C∗, extends to a CP1 family of Willmore surfaces f µ : M → S 4 where µ ∈ CP1 .