On a critical point theory for minimal surfaces spanning a wire in Rn.

In 1939 M. Morse and C. Tompkins [11] succeeded in extending the general theory of Morse (cp. e.g. [10]) on the relations between the number and types of critical points of a functional and topological properties of its level sets to the Plateau problem for minimal surfaces spanning a wire in IR. As a particular consequence of their results, from the existence of two minimal surfaces spanning a given wire which were disconnected in the set of all such minimal surfaces and of minimum type they were able to conclude the existence of a (third) minimal surface of non-minimum type. Independently and almost simultaneously the latter result was also obtained by M. Shiffman [18]. Later, in 1941, R. Courant [3] gave still a different proof using polygons to approximate a given boundary contour. (Cp. also [12], p. 379ff.) In all these papers minimal surfaces are characterized äs stationary points of the Dirichlet integral on the space H' n C°(5; IR} (cp. § 2 for notations), endowed with the norm of C°. The reason for choosing this topology is the need for "bounded compactness" of the space of admissible functions, which for the Plateau problem in the C°-norm is a consequence of the Courant-Lebesgue-Lemma.