The 6π Theorem About Minimal Surfaces

We prove that if Γ is a real-analytic Jordan curve in R 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 that asserts that all members of a geometrically defined class of curves cannot bound infinitely many minimal surfaces, stable or unstable.