Vertical Ends of Constant Mean Curvature H=1/2 in H^2\times R

Abstract

We prove a vertical halfspace theorem for surfaces with constant mean curvature $H=1/2,$ properly immersed in the product space $\h^2\times\re,$ where $\h^2$ is the hyperbolic plane and $\re$ is the set of real numbers. The proof is a geometric application of the classical maximum principle for second order elliptic PDE, using the family of non compact rotational $H=1/2$ surfaces in $\h^2\times\re.$

This is a revised version of the article that we submit before. There was a problem in the construction of graphical ends. We are presently working to fix it.The main geometric constructions will be mantained (replace the previous boundary with a planar boundary curve).Here we present the halfspace type theorem, that correspond to Section 4 of the previous article