All superconformal surfaces in ^4 in terms of minimal surfaces

We give an explicit construction of any simply-connected superconformal surface ϕ M²→ ⁴ in Euclidean space in terms of a pair of conjugate minimal surfaces g,h M²→⁴. That $ϕ$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in ⁴ and to images of superminimal surfaces in either a sphere ⁴ or a hyperbolic space ⁴ by an stereographic projection. We also determine the relation between the pairs $(g,h)$ of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in ⁴.