Global Properties of Minimal Surfaces in E 3 and E n

Abstract

A minimal surface is the surface of least area bounded by a given closed curve. In three dimensional space these surface are realized, for reasonably simple curves, by soap films spanning wire loops. This paper obtains results related to the theorem that the normals to a complete minimal surface, not a plane, are everywhere dense. Examples: l. The same theorem holds for two-dimensional surfaces in Euclidean spaces of dimension three or more. 2. Theorems connecting total curvature, Euler characteristic, and number of boundary components. 3. Theorems about the normals and tangent planes to minimal surfaces, including the capacity of the set of directions omitted. (Author)