The number of minimal surfaces bounded by Enneper's wire

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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 √ 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 x₁=0 and x₂ = 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 Γ bounding more than one minimal surface for which a specific bound on the number of minimal surfaces bounded by Γ is known.