New complete embedded minimal surfaces in $${{mathbb {H} ^2times mathbb {R}}}$$

We construct three kinds of complete embedded minimal surfaces in ({mathbb {H}^2times mathbb {R}}) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in ({mathbb {R}^3}) . The third one is a finite total curvature surface which is conformal to ({mathbb {S}^2setminus{p_1,ldots,p_k}, kgeq3.})