Classification of stable minimal surfaces bounded by Jordan curves in close planes in Euclidean three-space

In this thesis we study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in two close parallel planes in Euclidean three-space.;Our main result is to give a classification of these minimal surfaces, under certain natural asymptotic geometric constraints, in terms of certain associated varifolds which can be enumerated explicitly.;One consequence of the main theorem is that under our hypotheses, there exists a unique area-minimizing surface, and this surface has the largest possible genus among all stable embedded minimal surfaces with boundary the two families of curves introduced above.