Minimal surfaces in H × R

We construct complete embedded minimal surfaces in H × R. The first one is a finite total curvature surface which is conformal to S \ {p1, ..., pk}, k ≥ 2; the second one is a 1-parameter family of singly-periodic minimal surfaces which is asymptotic to a horizontal plane and a vertical plane; the third one is a 2-parameter family of minimal surfaces which have a fundamental piece of finite total curvature and is asymptotic to a finite number of vertical planes. We also show that a Scherk minimal graph over a domain in H bounded by an ideal rectangle is the only complete minimal surface of total curvature −2π.