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 ${\mathbb R^4}$.