Stabilität Mehrfach Zusammenhängender Minimalflächen

Abstract

Multiply connected minimal surfaces of genus 0 with only simple interior branch points, for which the corresponding boundary value problem

$$\Delta h - K|x_z |^2 h = 0; h_{|\partial \Omega } = 0$$

(K is the Gauss curvature and x_z is the complex gradient of the surface x) is uniquely solvable and which have the property, that the condition K|x_z|²≠0 holds in the branch points, are always isolated and stable solutions of the Plateau problem, corresponding to their boundary curves. To achieve these results one has to consider the conformal type as a variable. We give a method to perform the variation of the conformal type for holomorphic functions. Using the Weierstrass representation we thus obtain a differentiable structure on the set of multiply connected minimal surfaces. We find interesting connections between the classical Riemann-Hilbert problem and Fredholm properties of a projection operator on this manifold.