Preprint A561/2007
All superconformal surfaces in R^4 in terms of minimal surfaces

Ruy Tojeiro | Dajczer, Marcos

**Keywords: **

We give an explicit construction of any simply-connected superconformal surface
$\phi\colon\,M^2\to \R^4$ in Euclidean space
in terms of a pair of conjugate minimal surfaces $g,h\colon\,M^2\to\R^4$.
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 $\R^4$ and to images of superminimal surfaces in either a sphere $\Sf^4$ or a hyperbolic
space $\Hy^4$ 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 $\R^4$.