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In this paper, we consider minimal hypersurfaces in the product space$\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotationhypersurfaces and hypersurfaces invariant under hyperbolic translations. Wethen consider minimal hypersurfaces with finite total curvature. Thisassumption implies that the corresponding curvature goes to zero uniformly atinfinity. We show that surfaces with finite total intrinsic curvature havefinite index. The converse statement is not true as shown by our examples whichalso serve as useful barriers.

Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index