On minimal surfaces with free boundaries in given homotopy classes
Let S be a compact smooth embedded surface in \({\mathbb{R}}^ 3\) and let B be the unit disc in \({\mathbb{R}}^ 2\). The author considers the problem of finding a surface that minimizes the area (Dirichlet integral) among all surfaces having a conformal parametrization \(u: B\to u(B)\subset {\mathbb{R}}^ 3\) with boundary values \(u| \partial B\) in a given nontrivial homotopy class of curves \(\gamma\) : \(\partial B\to S\). He shows that this problem is solved whenever a certain pinching condition is satisfied. For homology classes, the corresponding result is due to \textit{N. Davids} [Am. J. Math. 64, 348-362 (1942)] who used a method by R. Courant. The author obtains this result also for homotopy classes because he keeps the lengths of the boundary curves of the minimizing sequence as short as possible. Thus he obtains the uniform continuity of these boundary curves with the aid of the Courant-Lebesgue-Lemma.
scientific article; zbMATH DE number 3921391