On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi - Dimensional Euclidean Spaces

An extension of the classic Enneper–Weierstrass representation for conformally pa-rametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP 1 and CP 2 sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3-and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of these sigma models are established. In particular, it is demonstrated that the generalised Weier-strass representation can admit different CMC-surfaces in R 3 which have globally the same Gauss map. A new procedure for constructing CMC-surfaces in R n is presented and illustrated in some explicit examples.