Communications in Analysis and Geometry

Volume 16 (2008)

Number 5

On complete mean curvature $\frac{1}{2}$ surfaces in $\HY ^2 \times \R$

Pages: 989 – 1005

DOI: http://dx.doi.org/10.4310/CAG.2008.v16.n5.a4

Authors

Laurent Hauswirth (Laboratoire d’Analyse et de Mathématiques Appliquées, Université Marne-la-Vallée)

Harold Rosenberg (Institut de Mathématique, 2 Place Jussieu, 75005 Paris, France)

Joel Spruck (Department of Mathematics, Johns Hopkins University)

Abstract

For a complete embedded surface with compact boundary andconstant mean curvature $\frac12$ in $\HY^2 \times\R$ lying on one side of a horocylinder, we prove ananalogue of the Hoffman-Meeks half-space theorem. As anapplication, we show that a complete immersed surface ofconstant mean curvature $\frac12$ which is transverse tothe vertical killing field must be an entire graph.Moreover, to each holomorphic quadratic differential on theunit disk or $\mathbb{C}$ we can associate an entire graphof constant mean curvature $\frac12$.

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