The number of minimal surfaces bounded by Enneper's wire

Abstract

Enneper's wire, the image of the circle of radius $R$ under Enneper's surface, bounds exactly three minimal surfaces for $R$ between 1 and $\sqrt 3$, and these three surfaces depend continuously on $R$. The other two surfaces (besides Enneper's surface) are absolute minima of area among disk-type surfaces bounded by Enneper's wire. These surfaces each have a unique horizontal tangent plane, whose height can be computed from $R$, and they are invariant under reflections in the planes $x_1=0$ and $x_2 = 0$. These two surfaces have positive second variation of area, and depend continuously on $R$. This result solves three open problems from the list in Nitche's 1989 book. Enneper's wire is the only Jordan curve $\Gamma$ bounding more than one minimal surface for which a specific bound on the number of minimal surfaces bounded by $\Gamma$ is known.