All superconformal surfaces in R-4 in terms of minimal surfaces

作者：Dajczer Marcos^*; Tojeiro Ruy

来源：Mathematische Zeitschrift, 2009, 261(4): 869-890.

DOI：10.1007/s00209-008-0355-0

We give an explicit construction of any simply connected superconformal surface face phi: M-2 -> R-4 in Euclidean space in terms of a pair of conjugate minimal surfaces g, h : M-2 -> 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 S-4 or a hyperbolic space 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 R-4.

• 出版日期2009-4