Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in ${\bf R}\sp n$. I. $n\geq 4$

Abstract

ABSTRACT.A degree theory applicable to Plateau's problem is developed and the Morse equality for minimal surfaces spanning a contour in Rn, n > 4, is proved.In [10] the author and David Elworthy developed a theory of degree for Fredholm maps on oriented Banach manifolds, a theory which generalized the now classical Leray-Schauder degree theory.The purpose of this paper is to show how one can use a slight extension of Elworthy-Tromba theory to develop a "Morse-theory" for minimal surfaces of disc type spanning a wire in Rn. We prove, via this theory, the first and last of Morse's "inequalities", and in so doing obtain an existence theorem to the classical problem of Plateau, and simultaneously a generalization of the famous mountain pass result for minimal surfaces due to Morse, Shiftman and Tompkins.Before proceeding with a detailed statement of the results we would like to present a bit of historical background.