Scholia

Abstract

The solution of Plateau’s problem presented by J. Douglas (1931) and T. Radó (1930) was achieved by a – very natural – redefinition of the notion of a minimal surface X:Ω→ℝ3 which is also used in our book: Such a surface is a harmonic and conformally parametrized mapping; but it is not assumed to be an immersion. Consequently X may possess branch points, and thus some authors speak of “branched immersions”. This raises the question whether or not Plateau’s problem always has a solution which is immersed, i.e. regular in the sense of differential geometry. Certainly there exist minimal surfaces with branch points; but one might conjecture that area minimizing solutions of Plateau’s problem are free of (interior) branch points. To be specific, let Γ be a closed, rectifiable Jordan curve in ℝ3, and denote by the class of disk-type surfaces X:B→ℝ3 bounded by Γ which was defined in Chap. 1. Then one may ask: Suppose that is a disk-type minimal surface \(X : \overline {B} \to \mathbb{R}^{3}\) which minimizes both A and D in . Does X have branch points in B (or in \(\overline {B}\) )? KeywordsMinimal SurfaceBranch PointJordan CurveHigh Order DerivativeUniformization TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.