How big is the family of stationary null scrolls?

Every minimal ruled surface in the Euclidean three space, E3E3, is congruent to either a plane or a helicoid. In particular, the plane is the only minimal cylinder in E3E3. An analogous result is known in the Lorentz–Minkowski three space, L3L3, if we restrict ourselves to those ruled surfaces where the ruling flow is non-null. In fact, the moduli space of stationary Lorentzian ruled surfaces with non-null ruling flow is formed by five congruence classes. However, when the ruled surfaces are generated by lightlike (null) ruling flows in L3L3, then a deep difference, in the study of cylinders seen as null scrolls, is obtained. We show that the moduli space of stationary null scrolls in L3L3 can be regarded as a kind of circle bundle on the moduli space of null curves in that background. This completes the classification of stationary ruled surfaces in L3L3.