Some recent developments in the theory of minimal surfaces

I suspect that each person can recall some moments from his childhood when he was captived by the beauty of soap bubbles. Their form can be truly marvelous. As adults, we do not tend to stare at soap bubble clusters in amazement, but as a professional mathematician, it happens that once one begins their study, one can remain seduced for a lifetime. My first contact with minimal surfaces happened some twenty years ago. I met Andre Haefliger while walking in Paris, we continued walking together and he asked me: ”Is there a foliation of R by minimal surfaces, other than a foliation by parallel planes?” This (apparently easy) question made me think about the nature of a minimal surface; I have not stopped thinking about it since then. We will see in section 7, that Haefligers’ question is unsolved for minimal laminations of R. Certainly soap films and soap bubbles seem a specialized subject; yet analysis, geometry, and topology are fundamental to their understanding. Soap bubble clusters are modelled on surfaces of constant mean curvature H , meeting along a curve S in a precise manner. At a smooth point of S, there are three smooth surfaces (of the cluster) meeting at equal angles (i.e., 120) and S has isolated singular points where six surfaces of the cluster meet at angles approximately 109. For example, the barycenter of a regular tetrahedron, and the triangles meeting at the barycenter which are formed by the edges of the tetrahedra and the barycenter. These properties of soap bubble clusters were observed by Plateau in the years 1870 and were established only relatively recently. Some excellent references for this subject are [28], [29], [33], [41]. We state a mathematical interpretation of soap bubbles. Consider positive real numbers V1, ..., Vn and surface configurations that separate space R 3 into regions having volumes V1, ..., Vn. A soap bubble is such a configuration such that the area of the surfaces in the cluster is a (local) minimum among all such surfaces bounding the volumes V1, ..., Vn. When