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Proceedings of the London Mathematical Society Advance Access originally published online on October 7, 2008
Proceedings of the London Mathematical Society 2009 98(2):445-470; doi:10.1112/plms/pdn038
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© 2008 London Mathematical Society

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

Benoît Daniel

Université Paris-Est
Laboratoire d’Analyse et de Mathématiques Appliquées
UMR 8050
UFR des Sciences et Technologies
61 avenue du Général de Gaulle
94010 Créteil cedex
France

Laurent Hauswirth

Université Paris-Est
Laboratoire d’Analyse et de Mathématiques Appliquées
UMR 8050
Cité Descartes, 5 bd Descartes
Champs-sur-Marne
77454 Marne-la-Vallée cedex 2
France
laurent.hauswirth@univ-mlv.fr

Received 14 December 2007. Revision received 3 June 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) 1/2 graph in H2 x R, and vice versa. This gives a classification of all entire CMC 1/2 graphs in H2 x R. Finally we construct properly embedded CMC 1/2 annuli in H2 x R.

2000 Mathematics Subject Classification 53A10, 53C42 (primary), 53A35, 53C43 (secondary).


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