The Riemann minimal examples

TL;DR

This paper reviews Riemann's discovery of a family of minimal surfaces with specific intersection properties and classifies all properly embedded minimal planar domains in three-dimensional space.

Contribution

It provides a complete outline of the recent proof classifying all properly embedded minimal planar domains as Riemann examples, catenoids, helicoids, or planes.

Findings

Riemann's minimal examples form a 1-parameter family with specific geometric limits.

Properly embedded minimal planar domains are classified into four types.

The paper revisits Riemann's original proof and presents a modern classification result.

Abstract

Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_{\lambda}$, $\lambda\in (0,\infty)$, of periodic properly embedded minimal surfaces in $\mathbb{R}^3$ with the property that every horizontal plane intersects each of his examples in either a circle or a straight line. Furthermore, as the parameter $\lambda\to 0$ his surfaces converge to a vertical catenoid and as $\lambda\to \infty$ his surfaces converge to a vertical helicoid. Since Riemann's minimal examples are topologically planar domains that are periodic with the fundamental domains for the associated $\mathbb{Z}$-action being diffeomorphic to a compact annulus punctured in a single point, then topologically each of these surfaces is diffeomorphic to the unique genus zero surface with two limit ends. Also he described his surfaces analytically in terms of elliptic functions on rectangular elliptic curves. This article exams Riemann's original proof of the classification of minimal surfaces foliated by circles and lines in parallel planes and presents a complete outline of the recent proof that every properly embedded minimal planar domain in $\mathbb{R}^3$ is either a Riemann minimal example, a catenoid, a helicoid or a plane.