Classification of Stable Minimal Surfaces Bounded by Jordan Curves in Close Planes

摘要

We study compact stable embedded minimal surfaces whose boundary is given bytwo collections of closed smooth Jordan curves in close planes of Euclidean3-space. Our main result is a classification of these minimal surfaces, undercertain natural geometric asymptotic constraints, in terms of certainassociated varifolds which can be enumerated explicitely. One consequence ofthis result is the uniqueness of the area minimizing examples. Another is theasymptotic nonexistence of stable compact embedded minimal surfaces of positivegenus bounded by two convex curves in parallel planes.