  We construct a one-parameter family of properly embedded minimal annuli in
the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric.
These annuli are not rotationally invariant. This family gives a vertical
half-space theorem and proves that each complete minimal graph in Nil_3 is
entire. Also, the sister surface of an entire minimal graph in Nil_3 is an
entire constant mean curvature 1/2 graph in H^2 x R, and conversely. This gives
a classification of all entire constant mean curvature 1/2 graphs in H^2 x R.
Finally we construct properly embedded constant mean curvature 1/2 annuli in
H^2 x R.
