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 (3)

• Conformal Geometry of Surfaces in the 4-Sphere and Quaternions
• Adjoint Transform of Willmore Surfaces in $n$-sphere
• The Gauss map of minimal surfaces in the Heisenberg group

Cited by in corpus (6)

• Natural Intrinsic Geometrical Symmetries
• Invariants and bonnet-type theorem for surfaces in $\r^4$
• Moebius geometry of three dimensional Wintgen ideal submanifolds in S^5
• Minimal Surfaces in the Four-Dimensional Euclidean Space
• Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space
• Quartic differentials and harmonic maps in conformal surface geometry