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 $G\subset \mathbb{R}^{3}$ 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 $H\in \left( 0,1/2\right)$, if $\Omega$ is included in a region of $\mathbb{ H}^{2}\times \left\{ 0\right\}$ 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 $\partial\Omega,$ which can also be less than $-1.$