The $$\mu $$-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 $$\mathbb { C}_*$$C∗-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called $$\mu $$μ-Darboux transforms. We show that a $$\mu $$μ-Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a $$\mu $$μ-Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in $$\mathbb { C}\mathbb { P}^3$$CP3 which is canonically associated to a minimal surface $$f_{p,q}$$fp,q in the right-associated family of f. Here we use an extension of the notion of the associated family $$f_{p,q}$$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 $$\mu =1$$μ=1. Moreover, the family of Willmore surfaces $$\mu $$μ-Darboux transforms, $$\mu \in \mathbb { C}_*$$μ∈C∗, extends to a $$\mathbb { C}\mathbb { P}^1$$CP1 family of Willmore surfaces $$f^\mu : M \rightarrow S^4$$fμ:M→S4 where $$\mu \in \mathbb { C}\mathbb { P}^1$$μ∈CP1.