I will talk about the existence and the non existence of minimal graphs in the Heisenberg space, with a given boundary. Moreover height and area estimates for such graphs are obtained.

How the shape of the ends of a minimal surface determines the surface itself? We discuss the problem for minimal surfaces with two ends in $\mathbb{H}^2\times\mathbb{R}$. Namely, we prove a Schoen type theorem for such surfaces.

We prove existence of graphs over exterior domains of $H^2$ with constant mean curvature $H=1/2$ in $H^2xR$, provided the boundary curve satisfies some geometric conditions. Furthermore, we prove a vertical halfspace theorem for surfaces with constant mean curvature $H=1/2$ properly immersed in $H^2xR$.