All superconformal surfaces in \R^4 in terms of minimal surfaces

Abstract

We give an explicit construction of any simply-connected superconformal surface $\varphi :M^2\to {\mathbb{R}}^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h:M^2\to {\mathbb{R}}^4$. That $\varphi$ 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 $\Sf^4$ 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 ${\mathbb{R}}^4$.

References in corpus (1)

• Adjoint Transform of Willmore Surfaces in $n$-sphere

Cited by in corpus (2)

• Natural Intrinsic Geometrical Symmetries
• Minimal Surfaces in the Four-Dimensional Euclidean Space