PROPERNESS OF ASSOCIATED MINIMAL SURFACES

For any open Riemann surface N and finite subset 3 subset of S-1 = {z is an element of C vertical bar vertical bar z vertical bar = 1}, there exist an infinite closed set 3(N) subset of S-1 containing 3 and a null holomorphic curve F = (F-j)(j) = 1,2,3 : N -> C-3 such that the map 2) : 3(N) x N -> R-2, 2) (v, P) = Re ((F-1, F-2)(P)), is proper. In particular, Re (vF) : N -> R-3 is a proper conformal minimal immersion properly projecting into R-2 equivalent to R-2 x {0} subset of R-3, for all v is an element of 3(N).