  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.
