Nick Harrison

The study of minimal surfaces dates back to the 18^th century when Lagrange looked for solutions to what is now called Plateau’s problem in \(\mathbb{R}^3\) — the problem of determining whether or not there exists a surface of least area bounded by a given closed curve in space. This problem finds its physical motivation in the behavior of soap films. Lagrange, and many others, attempted to answer this question using the calculus of variations which led to the minimal surfaces equation, a particular partial differential equation for which solutions are functions whose graphs are minimal surfaces, and hence solutions to Plateau’s problem. The task then became trying to show this PDE is solvable.

As it is the historical starting point, we will begin this talk with a discussion of this variational approach to explore and visualize the problem. But this method has important shortcomings, which prevented Lagrange from getting much further than showing planar surfaces are minimal. To remedy this, we will jump nearly 200 years later to the 1950s, when Ennio de Giorgi developed the theory of perimeters, which proved to be a key alternative perspective coming from measure theory to understanding Plateau’s problem, its solutions, and the extension to minimal hypersurfaces in \(\mathbb{R}^n\), which would require an appropriate definition of the area of \((n-1)\)−dimensional surfaces and other generalizations.

No knowledge of measure theory will be assumed from the audience, so a mostly intuitive discussion of the perimeter will be given. In this new setting, we will be able to not only answer the question of existence of solutions, but to go further and explore the geometry of solutions, namely the regularity (i.e. smoothness) of these surfaces, which involves some very surprising results.