We prove that if Γ is a
real-analytic Jordan curve in R3 whose total curvature does not exceed 6π, then Γ
cannot bound infinitely many minimal surfaces of the topological type of the
disk. This generalizes an earlier theorem of J. C. C. Nitsche, who proved the
same conclusion under the additional hypothesis that Γ does not bound any
minimal surface with a branch point. It should be emphasized that the theorem
refers to arbitrary minimal surfaces, stable or unstable. This is the only
known theorem which asserts that all members of a geometrically defined
class of curves cannot bound infinitely many minimal surfaces, stable or
unstable.
