On the Morse number of embedded and non-embedded minimal immersions spanning wires on the boundary of special bodies in ?3

Abstract

It is now more than fifty five years since the first paper on what is now called "Morse theory" first appeared. Since that time it has had an important impact on both topology and analysis (e.g. see the recent survey article on Morse's work by Raoul Bott in the Bull. AMS [5]). One of Morse's main goals was to apply his ideas to the Calculus of Variations in several variables and in particular to the classical problem of Plateau. This attempt was undertaken jointly by Morse and C.B. Tompkins and also independently by Max Shiffman and Richard Courant. These people had some limited success. In particular Morse and Tompkins [16] and Shiffman [-24] were able to prove that for a given wire F which admits two disc type minimal surfaces which span it, which are isolated as minimal surfaces (in some weak topology) and which are strictly (locally) area minimizing then there exists a third minimal surface spanning F which cannot be an isolated strict minimum for area.This result would, given the nature of the problem, be a type of Morse inequality. However no one could extend these methods to the situation where one knew one had two minima in a fine topology (e.g. Sobolev H S or C k'~ as opposed to M-S-TC~ Furthermore if you had two minima which are embedded (non-embedded immersions) is the third surface an embedding (nonembedded immersion)? The methods employed by the early pioneers of Analysis in the Large break down completely in attempting to answer these questions, as does those of more modern approaches to Analysis in the Large by Palais and Smale [22,26].In [4] the author and Reinhold BShme developed a global index and stability theory of minimal surfaces of disc type. It was here that notions of non-degeneracy of minimal surfaces as critical points of an energy functional were introduced and where generic non-degeneracy and generic finiteness results for minimal surfaces of disc type were proved. This was the first concrete information on the number and isolated nature of solutions to Plateau's problem to be optained, and the first necessary step in the development of any Morse theory for Plateau's problem.Using this theory as a starting point the author was able to develop [-34, 150 A. The purpose of this paper is to employ these ideas to show that under certain circumstances one can actually count the Morse number of embeddings and immersions which are not embeddings spanning a given wire F. Statement of Main ResultLet D be the unit disc in N~, (?D=S 1 and c~: S1--+IR ", n>4 an embedding of Sobolev class H ~, with F~=c~(S 1) the image wire. Let d(c~) be the space of all H ~ maps, s>s/2, r>2s+l, u: D~N", u=(u 1 .... ,u") such that for all i, Au~=O (each u ~ is harmonic) and u: S1--+F ~ is homotopic to ~.A minimal surface of disc type is an element u~X(~) satisfying and E~ is equivariant with respect to this action. Consequently if u is any critical point of E= the orbit of N through a (9,(N) consists of critical points. For such a critical point let D2E(u): T~.X(e)x T,Y(:<)--+R denote the Hessianor second deri...