Some strange properties of minimal surfaces in connection with Plateau's problem
From the geometrical point of view, a minimal surface is defined by the fact that in every point the mean curvature is null. In the sense of \textit{L. Bianchi} [`Lezioni di geometria diferenziale', 2nd ed., Vol. 1 (Enrico Spoerri, Pisa) (1902; JFM 33.0633.01), see also JFM 18.0648.01, JFM 41.0676.01, JFM 48.0784.02, JFM 49.0498.06], Plateau's problem consists in finding a portion of a continuous minimal surface without interior singular points, terminated by a given closed curve. The minimal surface locally minimizes (or maximizes) the area among all surfaces passing through a local close contour lying on the minimal surface, which can be generated in a certain spatial vicinity of the minimal surface. We develop a method to generate continuous families of minimal surfaces passing through a given, fixed curve and we demonstrate the existence, at least for large classes of curves and boundary conditions. In the case of a closed curve, if these minimal surfaces have not interior singular points, they are solutions of Plateau's problem. For a class of curves still large enough, in which the parametric coordinates of the curve are given by trigonometric polynomials, we bring into evidence the exotic property of `cutting' of some surfaces, after a continuous zone lying in the vicinity of the curve. After this zone, the surface is `broken' in a number of `petals' evolving whimsically in space. We complete this special study with a package of programs for graphics and numerical calculations, necessary, on the one hand, to construct, to see, and to rotate the studied surfaces, and, on the other hand, to solve in a numerical way the equations of `cuts', which, as we shall see, cannot be solved directly except for the simplest cases. These programs were written in \texttt{FORTRAN}, calling, however, subroutines and functions of the \(C\) language (especially for graphics).
scientific article; zbMATH DE number 1087088