Morse-Conley theory for minimal surfaces of varying topological type

Abstract

After the Plateau problem had been solved by Douglas and Rad6 by finding a possibly branched disk type minimal immersion bounded by a given Jordan curve in ~N, soon more general problems were proposed. On one hand, the existence of minimal surfaces of higher topological structure was investigated by Douglas I-D], Courant [C], and Shiffman [Sh 1]. On the other hand, the problem of finding unstable minimal surfaces, and, more generally, developing a Morse theory for minimal surfaces of disk type was attacked by Morse-Tompkins [MT1] and Shiffman [Sh2]. A natural question then was to develop a Morse theory for minimal surfaces of arbitrary topological structure. While Morse-Tompkins in their paper [MT2] only treated a very special case, namely annulus type surfaces, of what their title "Unstable minimal surfaces of higher topological structure" claims, Shiffman confronted the case of genus 0 and arbitrary connectivity [Sh3]. These classical papers were not in every respect satisfactory. The investigations of Douglas and Courant on the higher genus problem were recently critisized by Tromba [T 1]; however, Luckhaus I-L] was able to carry out a systematic reworking of the arguments of Courant and Shiffman. The original approaches of Morse-Tompkins, Shiffman moreover severely suffer from the fact that these authors work in the C~ instead of the more natural H l"2-topology. In the C~ Dirichlet's integral (whose critical points parametrize the sought-after minimal surfaces) is not differentiable and no intrinsic notion (i.e. depending only on the surface) of Morse index can be defined. This makes it impossible to decide in the work of Morse-Tompkins and Shiffman whether the Morse relations for minimal surfaces reflect a property of the surfaces