Uniqueness of minimal surfaces whose boundary is a horizontal graph and some Bernstein problems in {{\mathbb{H}^{2}\times \mathbb{R}}}

We deduce that a connected compact immersed minimal surface in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ whose boundary has an injective horizontal projection on an admissible convex curve in ${\partial_\infty{\mathbb{H}^{2}\times \mathbb{R}}}$ , and satisfies an admissible bounded slope condition, is the Morrey’s solution of the Plateau problem and is a horizontal minimal graph. We prove that there is no entire horizontal minimal graph in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ .