Properly immersed minimal disks bounded by straight lines

Let pi (1) and pi (2) be two distinct parallel planes in R-3. Let omicron ( 1) epsilon pi (1) and omicron (2) epsilon pi (2) denote two points such tha t the segment l(0) = [omicron (1), omicron (2)] meets pi (1) and pi (2) ort hogonally. Let l(1) subset of pi (1) be a straight line containing omicron (1), and denote L as the set of straight lines in pi (2) containing omicron (2). Then there exists an analytic family {Y-theta : D-theta -->, R-3 : th eta epsilon [0, pi]} of proper pairwise non congruent minimal immersions sa tisfying: 1. D-theta is homeomorphic to (D(0, 1) over bar - {P-1, Q(1)), where {P-1, Q(1)} subset of S-1 = partial derivative(D(0, 1) over bar. 2. Y-theta(parti al derivativeD(theta)) = l(1) boolean OR l(0) boolean OR l(2), where l(2) e psilon L. 3. Y-theta(D-theta) is contained in the slab determined by pi (1) and pi (2). 4. If c(1) and c(2) are the two connected components of partia l derivativeD(theta), then Y-theta \(D theta -ci) is injective, i = 1, 2. 5 . The parameter theta is an analytic determination of the angle that the or thogonal projection of l(1) on pi (2) makes with l(2), and Y-theta(D-theta) is invariant under the reflection around a straight line not contained in the surface. 6. If Y : D --> R-3 is a proper minimal immersion satisfying 1 , 2, 3 and 4, then, up to a rigid motion, Y = Y-theta, theta epsilon [0, pi ].