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 $\overline {B}$ )?