Influence of boundary conditions on the existence and stability of minimal surfaces of revolution made of soap films

Abstract

Because of surface tension, soap films seek the shape that minimizes their surface energy and thus their surface area. This mathematical postulate allows one to predict the existence and stability of simple minimal surfaces. After briefly recalling classical results obtained in the case of symmetric catenoids that span two circular rings with the same radius, we discuss the role of boundary conditions on such shapes, working with two rings having different radii. We then investigate the conditions of existence and stability of other shapes that include two portions of catenoids connected by a planar soap film and half-symmetric catenoids for which we introduce a method of observation. We report a variety of experimental results including metastability—an hysteretic evolution of the shape taken by a soap film—explained using simple physical arguments. Working by analogy with the theory of phase transitions, we conclude by discussing universal behaviors of the studied minimal surfaces in the vicinity of their existence thresholds.