Extreme curves bound embedded minimal surfaces of the type of the disc

Abstract

In [3] Gulliver and Spruck proved that any extreme Jordan curve F whose total curvature does not exceed 4~z and which lies on the boundary of a convex set bounds a unique topologically embedded minimal surface of the type of the disc. If F is C a, ~ then this unique surface is a differentiable C a," embedding of the unit disc in JR. 2 into IRa [5]. In this paper we apply some of the global methods developed in [7] to give a direct and easy proof that the set of immersed minimal surfaces can be viewed as a differentiable submanifold of a product bundle over a space of immersed curves and that the bundle projection map when restricted to this submanifold is Fredholm of index zero. Using this fact, regularity results of minimal surfaces, some of the work of Gulliver and Spruck, and elementary transversality arguments we prove that any smooth curve on the boundary of a C 2 compact convex body in 1R 3 bounds a differentiably embedded disc. It is obvious that our hypotheses can be weakened in several directions, however the conceptual simplicity along with the absence of overbearing technical details justifies these hypotheses.