Minimal surface between two non coaxial rings

Question

I'm currently studying minimal surfaces using the Euler-Lagrange equation. I'm particularly interested in minimal surfaces between two circles.

I have already examined the case of two coaxial circles but I'm not able to find any literature on the more general case where the two circles are not coaxial. Does anyone know a good source of material for this problem?

The only thing I have found was Isenberg: The Science of Soap Films and Soap Bubbles, which only states the result of the case of two non-coaxial rings, omitting any further explanation.

Thank you very much!

Answer 1

You are looking for Riemann's minimal surfaces. This is a one-parameter of surfaces which includes the catenoid as a special case. The wikipedia page links to some nice images. Riemann's (posthumous) paper that introduces these surfaces is also available: Ueber die Fläche vom kleinsten Inhalt bei gegebener Begrenzung.

There are a number of articles related to these surfaces; unfortunately the search is a bit difficult because "Riemann surface" means something else. Often, these surfaces are called "Riemann examples" or "Riemann minimal examples". In an interview, Minicozzi mentioned them as a source of inspiration for the well-known series of papers by Colding and Minicozzi on the structure of minimal surfaces in three dimensions. Or at least this is how it was presented in the news item:

Using software that draws complex surfaces, they input a set of parameters for what's known as a Riemann shape, and a computer in Berkeley crunched the math and began to draw the figure. As Minicozzi gazed at the monitor of his own computer in his Krieger School office, the Riemann shape began to build out of helicoids. There was no mistaking what he was seeing. The spiral-staircase forms were the building blocks not only of the minimal disk, but of this complex surface, too.

"It was completely shocking," he recalls. "We hadn't known it would do that. All of a sudden, there it is. It was a good moment."