A classical example of Riemann of a one-parameter family of singly-periodic genus 0 minimal surfaces \(R_t\) satisfies the following properties: (1) every horizontal plane intersects \(R_t\) in a single component that is a circle or a straight line; (2) \(R_t\) is invariant under reflection in the \((x_1,x_3)\)-plane; (3) \(R_t\) is invariant under translation by \(v_t=(0,t,1)\); (4) far away from the line passing through the origin and in the direction \(v_t\), \(R_t\) is asymptotic to the family of parallel horizontal planes at integer heights.
The main result of the authors asserts that every properly imbedded minimal surface \(M\) with more than one end has asymptotic behavior that mimics the behavior of the examples given by Riemann. More precisely, if \(M\) has as limit tangent plane the \((x_1,x_2)\)-plane then there is a natural geometric ordering of the ends of \(M\) that is equivalent to the ordering of a compact subset of \([0,1]\). A topological uniqueness theorem is also proved for such surfaces.