Nonproper Complete Minimal Surfaces Embedded in ℍ2 × ℝ

Examples of complete minimal surfaces properly embedded in ${\BBH}^2\times {\BBR}$ have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper, we construct a large class of examples of complete minimal surfaces embedded in ${\BBH}^2\times {\BBR}$, not necessarily proper, which are invariant by a vertical translation or by a screw motion. In particular, we construct a large family of nonproper complete minimal disks embedded in ${\BBH}^2\times {\BBR}$ invariant by a vertical translation and a screw motion and whose importance is two-fold. They have finite total curvature in the quotient of ${\BBH}^2\times {\BBR}$ by the isometry, thus highlighting a very different behavior from minimal surfaces embedded in ${\BBR}^3$ satisfying the same properties. They show that the Calabi–Yau conjectures for embedded minimal surfaces do not hold in ${\BBH}^2\times {\BBR}$.