On July 20, 1866, three months before his death, Bernhard Riemann handed a few sheets of paper with formulas to Karl Hattendorff, one of his colleagues in Göttingen. Hattendorff worked out the details, and published this as a posthumous paper of Riemann. It contains Riemann’s work on minimal surfaces.
One of the problems Riemann addresses is to find all minimal surfaces with circular horizontal cross sections. A priori this is an overdetermined problem: Given two non-concentric circles in parallel planes, there is no reason to assume that a minimal surface having these circles as boundary would contain any further circles.
That Riemann’s intuition was justified follows a posteriori from a theorem by Max Shiffman from 1956: Any minimal annulus bounding two circles in horizontal planes has circles as all its horizontal cross sections.
Riemann obtains a 1-parameter family of solutions to his problem in terms of elliptic integrals. Curiously, it has the catenoid and helicoid as its limits. The helicoidal limit is the first example of a parking garage structure with two limit helicoids of opposite spin. It would be interesting to classify all singly periodic minimal surfaces with planar ends that have such parking garage structures as limits.
In 2015, Bill Meeks, Joaquin Pérez, and Antonio Ros proved that any properly embedded minimal surface of genus 0 is the plane, the catenoid, the helicoid, or one of Riemann’s examples.
I doubt, however, that Riemann’s minimal surface has revealed all its mysteries.
Resources
B. Riemann: Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), 3–52
M. Shiffman: On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. 63 (1956), 77–90
W. H. Meeks III, J. Pérez, A. Ros: Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998),107–132
W. H. Meeks III, J. Pérez, A. Ros: Properly embedded minimal planar domains,
Ann. of Math. 181 (2015), 473-546
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 