ABSTRACT

Prior to the mathematical study of minimal surfaces, we give a brief description of their fundamental properties which can be demonstrated in the language of visual geometry. When the Belgian physicist, professor of physics and anatomy Joseph Plateau (1891–1883), started experimenting with soap films to examine their configurations, he could hardly believe that it would be the beginning of an important branch in science still developing rapidly and known today under the general name of the “Plateau problem”. Some experiments carried out by Plateau are very simple and well-known to the reader: Probably, everyone amused oneself by blowing soap bubbles through a pipe or by constructing various soap films spanning a wire frame. It is well known that if a closed wire frame (e.g., homeomorphic to a circle) is dipped into soapy water and then taken out carefully, a beautiful iridescent soap film hangs from it. The film dimensions may be considerable; though the larger the film is, the sooner and easier it bursts due to gravity. If, on the contrary, the film dimensions are relatively small, then gravity may be neglected when some important properties of soap films are studied. This circumstance will be constantly used in the sequel. Minimal surfaces are a mathematical object modeling physical soap films fairly well. Conversely, many important properties of minimal surfaces are explicit in simple physical experiments with soap films. There are many branches of mathematics, which arose from concrete physical and applied problems. However, not all of them are as closely as the Plateau problem related to so many mathematical theories different in machinery. Within the framework of the theory of minimal surfaces, Lie groups, homology and cohomology theories, bordisms, etc., are interlinked.