The classical plateau problem and the topology of three-dimensional manifolds: The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's Lemma

Abstract

LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6[45] (independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman[42], who proved that there are no interior “true” branch points, and by Gulliver[lS], who proved that there are no interior “false” branch points. In 1948, Morrey[35] devised a new method to solve the Plateau problem for a map from the disk into a “homogeneously regular*’ Riemannian manifold. Moreover, he proved the interior regularity of the map in case the ambient manifold is regular, and that the map is real analytic if the ambient manifold is real analytic. The arguments of Osserman and Gulliver in addition show that Morrey’s solution has no interior branch point when the ambient manifold is three-dimensional. In 1951, Lewy[29] showed that if the Jordan curve y is also real analytic, in a real analytic manifold, then any minimal surface with boundary y is real analytic up to the boundary. Hence in this case Morrey’s map is real analytic map on the closed disk. (For a proof of Lewy’s Theorem in a general real analytic manifold, see HiIdebrandt[23].) In 1969, Hildebrandt[23] proved that the Douglas solution is smooth up to the boundary if the Jordan curve is smooth and regular. (Further improvements are due to Kinderlehrer [25], Nitsche [40], and Warschawski[55].) In [22], Heinz and Hildebrandt extended Hildebrandt’s result to minimal surfaces in general Riemannian manifolds. Once we have boundary regularity, it makes sense to ask whether Douglas’ or Morrey’s solution of the Plateau problem has a boundary branch point or not, To date, this problem has not been settled. The first partial result in this direction is due to Nitsche[40], who showed that there are only a finite number of boundary branch points for minimal surfaces with smooth boundary in R3. This was then generalized by Heinz and Hildebrandt to smooth manifolds. Gulliver and Lesley[l6] have also observed that the Douglas-Morrey solution for a real analytic curve in a real analytic manifold has no boundary branch point, using the previously mentioned result of Lewy. Despite all these results, an interesting topological question remained unsolved, namely, under what conditions is the Douglas solution an embedded surface? It has been generally conjectured that when the Jordan curve is extremal, i.e. lies on the