We study the existence and uniqueness problem of compact minimal vertical graphs in $\mathbb{ H}^n\times\mathbb{ R}$, $n\geq 2$, over bounded domains in the slice $\mathbb{ H}^n\times\{ 0\}$, with non-connected boundary having a finite number of $C^0$ hypersufaces homeomorphic to the sphere $\mathbb{ S}^{ n-1}$, with prescribed bounded continuous boundary data, under hypotheses relating those data and the geometry of the boundary. We show the nonexistence of compact minimal vertical graphs in $\mathbb{ H}^n\times\mathbb{ R}$ having the boundary in two slices and the height greater than or equal to $\pi/(2n-2)$.
Categories: math.DG