Almost every curve in R3 bounds a unique area minimizing surface

Abstract

Curiously enough, such a surface need not be unique. Examples of this nonuniqueness abound. Nitsche [17] thoroughly develops a family of examples by taking intersections of Enneper's minimal surface (Fig. 1) with ellipsoids x 2 +y2 +~z2 = a 2. (See Fig. 2.) The intersection with small ellipsoids is nearly planar (Fig. 2b), and the enclosed portion of Enneper's surface gives the unique area minimizing surface. As the ellipsoids become larger, eventually (Fig.2f) the enclosed portion of Enneper's surface is no longer area minimizing and there are at least two different area minimizing surfaces, which presumably look like Figure 3. This example provides a one parameter family of nonsimilar curves bounding more than one area minimizing surface. By symmetrically adding small, smooth bumps to these examples, one can obtain a large space of curves bounding more than one area minimizing surface, a space in some sense of the same dimension as the entire space of curves we shall consider in this paper. Nevertheless, we shall prove that the probability of picking such a curve at random is zero. Nitsche [16, pp. 396-398] refers to many other examples. The author [13] gives an example of an analytic curve in R 4 that bounds a whole continuum of distinct area minimizing surfaces. (See also Fleming [9], L6vy [12, p.29], Courant [5, pp. 119-122].)