Deceiving Simplicity (Annuli VI)

Just three months before his death on July 20, 1866 (150 years ago), Bernhard Riemann handed a few sheets of paper with formulas to Karl Hattendorff, one of his colleagues in Göttingen.
Hattendorff did better than Riemann’s house keeper who discarded the papers and notes she found.

He instead worked out the details, and published this as a posthumous paper of Riemann. It contains his work on minimal surfaces. Riemann was possibly the first person who realized that the Gauss map of a minimal surface is conformal, and that its inverse is well suited to find explicit parametrizations. He used this insight to construct the minimal surface family that bears his name, as well as a few others that were later rediscovered by Hermann Schwarz.

Above is one of Riemann’s minimal surfaces, parametrized by the inverse of the Gauss map. This means in particular that the surface normal along the parameter lines traces out great circles on the sphere. Riemann discovered these surfaces by classifying all minimal surfaces whose intersections with horizontal planes are lines or circles. These are the catenoid, the helicoid, or Riemann’s new 1-parameter family.

The proof utilizes elliptic functions, which is not surprising: Riemann’s minimal surfaces are translation invariant, and their quotient by this translation is a torus, on which the Gauss map is a meromorphic function of degree 2. It is in fact one of the simplest elliptic functions, and one can use it to parametrize Riemann’s surfaces quite elegantly. What is not simple is the proof that these surfaces have indeed circles as horizontal slices. All arguments I know involve some more or less heavy computation. We are clearly lacking some insight here.

The longer one studies these surfaces, the more perplexing they become. There is, for instance, Max Shiffman’s theorem from 1956. It states that if a minimal cylinder has just two horizontal circular slices, all its horizontal slices are circles. The proof is elegant, magical, and still mysterious, just like Riemann’s minimal surfaces.