Half-space Theorem, Embedded Minimal Annuli and Minimal Graphs in the Heisenberg Group

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil3 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 Nil3 is entire. Also, the sister surface of an entire minimal graph in Nil3 is an entire constant mean curvature 1 2 graph in H × R, and conversely. This gives a classification of all entire constant mean curvature 1 2 graphs in H×R. Finally we construct properly embedded constant mean curvature 1 2 annuli in H × R.