On the structure of minimum surfaces at the boundary

Graduation date: 1963In this paper we consider the behavior of certain surfaces at\ud certain boundary points. The surfaces under consideration satisfy\ud a topological definition and are of 2-dimension in 3-dimensional\ud Euclidean space with the boundary a finite set of straight line\ud segments. It is shown that the surface of minimum area with a given\ud boundary is locally Euclidean at all non-vertex. boundary points.\ud The key to the proof is a theorem in 1 which itself concerns the\ud behavior of a set of points under very restricted conditions. It\ud is shown in 1 that for almost all interior points the conditions\ud of the lemma are satisfied.\ud This paper first shows that the part of the given surface\ud interior to some sphere centered at any non-vertex boundary point\ud lies near a plane passing through the point in question. Secondly\ud it is shown that for any point of the surface lying in the sphere\ud the surface interior to any smaller neighborhood of that point lies\ud near a plane. "Near" here refers to nearness with respect to the\ud radius of the sphere or neighborhood in consideration.\ud From these conditions we may construct a bounded point set\ud satisfying the hypothesis of the aforementioned theorem. The\ud theorem of this paper follows immediately