CMC Graphs With Planar Boundary in $mathbb{H}^{2}times mathbb{R}$

It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included in a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$