On the existence of higher dimensional Enneper's surface
The author shows the existence of complete minimal immersions of \(\mathbb{R}^n\) into \(\mathbb{R}^{n+1}\) for \(n=3,4,5,6\), which are generalizations of the classical Enneper's minimal surface in \(\mathbb{R}^3\). His proof consists of three steps. Firstly, he solves a Dirichlet problem for the minimal surface equation and gets a certain \(n\)-dimensional minimal hypersurface in \(\mathbb{R}^{n+1}\). Secondly, he obtains a compact Enneper type surface by reflecting the minimal hypersurface across the totally geodesic part of its boundary. Thirdly, by blowing up the compact Enneper type surface he constructs the desired complete minimal hypersurface in \(\mathbb{R}^{n+1}\). The process of blowing up requires some curvature estimates. For this reason, the construction is restricted to the case \(n\leq 6\).
scientific article; zbMATH DE number 1008569