Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

Abstract : 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 of annuli is used to prove a vertical half-space theorem which is then applied to prove that each complete minimal graph in Nil(3) is entire. Also, it is shown that the sister surface of an entire minimal graph in Nil(3) is an entire constant mean curvature (CMC) 1/2 graph in H(2) x R, and vice versa. This gives a classification of all entire CMC 1/2 graphs in H(2) x R. Finally we construct properly embedded CMC 1/2 annuli in H(2) x R.