On complete mean curvature 1/2 surfaces in HXR

Laurent Hauswirth (1) , Harold Rosenberg (2) , Joel Spruck

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 IMJ - Institut de Mathématiques de Jussieu

We prove an half-space theorem for Constant mean curvature surfaces H=1/2 in H(2)xR. Let S be a properly embedded constant mean curvature 1 /2 surface in H(2)xR. Suppose S is asymptotic to a horocylinder C, and on onevside of C. If the mean curvature vector of S has the same direction as that of C at points of S converging to C, then S is equal to C (or a subset of C if the boundary of S is empty). Next we apply this theorem to classify complete graph. Let S be a complete immersed surface in H(2)xR of constant mean curvature H = 1/2. If S is transverse to Z then S is an entire vertical graph over H(2).

Domaines: Géométrie différentielle [math.DG]

Laurent Hauswirth, Harold Rosenberg, Joel Spruck. On complete mean curvature 1/2 surfaces in HXR. Comunication in Analysis and Geometry, 2008, 98 (3), pp.445-470. ⟨hal-00796851⟩