The existence of embedded minimal surfaces and the problem of uniqueness

Abstract

It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution. (Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a classical surface. However, the Douglas-Radd-Morrey solution may in general have selfintersections. It was then a classical problem to find conditions on the Jordan curve to guarantee that the solution is an embedded disk. In 1932 Radd was the first to produce a general condition of this type. He proved that if the curve can be projected onto a convex curve in the plane, then it bounds a unique embedded minimal disk which is a graph over this plane. It was not until the seventies that there was further progress on the problem of embeddedness. Osserman had conjectured that if a Jordan curve is extreme, that is, lies on the boundary of its convex hull, then the Douglas-Rad6 solution is embedded. Gulliver and Spruck [10] proved this conjecture under the additional assumption that the absolute total curvature of the Jordan curve is less than 4~z. Right after this work, Tomi and Tromba [32] showed that an extreme curve must bound an embedded minimal disk which need not be stable. In 1976 Almgren and Simon [1] established the existence for such a curve of an embedded stable minimal disk (which need not be a Douglas-Radd solution). Finally in the same year, the authors [18] proved that any Douglas-Rad6 solution must be embedded.