Classification of Stable Minimal Surfaces Bounded by Jordan Curves in Close Planes
by
Rosanna Galotta
University of Massachusetts

This paper exploits the behavior of stable minimal surfaces bounded by a finite number of disjoint compact Jordan curves in two close planes of Euclidean 3-space as one of the two planes approaches the other. Here ``close planes'' means:either planes which form a small positive angle, or parallel planes such that the distance between them is small. In the first case, the planes approach each other by a (small) rotation, while in the second case they approach each other by a (small) translation orthogonal to the planes. A partial description of such minimal surfaces was known, under the hypotheses of their existence and area-minimality. Here we first show that, if the two planes are close enough, there always exists a stable minimal surface bounded by the given Jordan curves whose rotation or orthogonal projections on any of the two planes intersect transversally in a finite number of points, called ``crossing points''. We see that the surfaces are made up by almost flat graphs away from the crossing points, and by almost helicoidal pieces or union of two flat graphs near the crossing points, depending on the multiplicity of the limiting varifold. We then consider the question of uniqueness, and show that the topology provides an upper bound on the number of stable minimal surfaces having the same boundary and the same number of ``helicoidal'' crossing points and of ``flat'' crossing points, and that the area-minimizing examples are unique. Moreover we prove some interesting non-existence results for stable and unstable minimal surfaces bounded by two convex Jordan curves in close planes. Such a study is in the spirit of a problem posed by Tibor Rado in 1946 at the Princeton Bicentennial Conference, to estimate the number of minimal surfaces bounded by one or more given curves in Euclidean three-space.

Date received: February 23, 1998