All superconformal surfaces in $${\mathbb R^4}$$ in terms of minimal surfaces

Abstract

We give an explicit construction of any simply connected superconformal surface \({\phi:\,M^2\to \mathbb R^4}\) in Euclidean space in terms of a pair of conjugate minimal surfaces \({g,h:\,M^2\to\mathbb R^4}\). That \({\phi}\) 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 \({\mathbb R^4}\) and to images of superminimal surfaces in either a sphere \({\mathbb S^4}\) or a hyperbolic space \({\mathbb H^4}\) 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 \({\mathbb R^4}\).