The Problem of Plateau on a Riemannian Manifold

The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas 25' and Rado 15. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 38 on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward direct methods of the Calculus of Variations (see 30). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface (5, 19, 27, 28, 129], 33, 135], 41, 43), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see 18, 22, 23, 24). Finally minimal surfaces which do not furnish even relative minima have been discussed (see 20, 21, 31, 32, 33, 34, 35, 39, 40, 42, 43). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth 7 where the space considered is hyperbolic space. However, S. Bochner 2 has studied the harmonic functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is harmonic in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a homogeneously (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo-