Fig. 3.

The Riemann examples are a two-parameter family of periodic complete minimal surfaces, where the parameters can be thought of as the size of the necks and the angle from one fundamental domain to the next. By choosing the two parameters appropriately, one can produce sequences of Riemann examples that illustrate both structure theorems. As the parameters degenerate, the Riemann examples look like either a collection of catenoids stacked on top of each other or two oppositely oriented helicoids (with parallel axes) glued together. Figs. 3–6 show various stages of degeneration. The first two (Figs. 3 and 4) are where the necks are far apart and the surface looks like a collection of catenoids stacked on top of each other; in Fig. 5, the necks have moved closer to each other, and in Fig. 6, the necks are almost on top of each other, and the surface looks like two oppositely oriented helicoids glued together.