Limit Surfaces of Riemann Examples

The only connected minimal surfaces foliated by circles and lines are domains on one of the following surfaces: the helicoid, the catenoid, the plane, and the examples of Riemann ([Ri] p329-33, [En] p403-6, [Ni] p85-6). All these surfaces are complete and embedded. Topologically they are planar domains: the helicoid is simply-connected, the catenoid is an annulus (conformally a twice-punctured sphere), and each Riemann example (see Figure 1) is conformal to the plane minus the points {(n, 0), ( 1 n , 0) |n ∈ Z} [HKR]. In this section we will show that the plane, helicoid, and catenoid arise naturally as limits of well-chosen and properly normalized sequences of Riemann examples. The local behavior of domains on Riemann examples accounts for the existence of these limits, thus allowing a change of topology in the limit surfaces.