Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group
Benôıt Daniel, Laurent Hauswirth · Proceedings of the London Mathematical Society · 2008
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) ½ graph in ℍ2 × ℝ, and vice versa. This gives a classification of all entire CMC ½ graphs in ℍ2 × ℝ. Finally we construct properly embedded CMC ½ annuli in ℍ2 × ℝ.