12.  Applications to geometry

12.1  Plateau problem

The classical Plateau's problem in three dimensions consists in showing that, given an edge of an unknown surface given by a prescribed simple and closed boundary curve, there exists a minimal surface resting on this edge. It was posed in 1760 by the Italian mathematician, mechanic and astronomer and naturalized Frenchman J.-L. Lagrange (1736-1813), but is named after the Belgian physicist and mathematician J. Plateau (1801-1883), who was interested in soap films (1873).

A minimal surface is one that locally minimizes area, or equivalently has zero mean curvature.

A catenoid (figure 13 ) is a minimal surface lying between...