Minimal cylinders in the three-dimensional Heisenberg group

We study minimal cylinders in the three-dimensional Heisenberg group Nil_3 using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders in the three-dimensional Minkowski space can also be obtained.