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The Riemann minimal examples

Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_λ$, $λ\in (0,\infty)$, 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 $λ\to 0$ his surfaces converge to a vertical catenoid and as $λ\to \infty$ 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.