  We study compact stable embedded minimal surfaces whose boundary is given by
two collections of closed smooth Jordan curves in close planes of Euclidean
3-space. Our main result is a classification of these minimal surfaces, under
certain natural geometric asymptotic constraints, in terms of certain
associated varifolds which can be enumerated explicitely. One consequence of
this result is the uniqueness of the area minimizing examples. Another is the
asymptotic nonexistence of stable compact embedded minimal surfaces of positive
genus bounded by two convex curves in parallel planes.
