SADDLE TOWERS AND MINIMAL k-NOIDS IN H2 x R

Given k >= 2, we construct a (2k - 2)-parameter family of properly embedded minimal surfaces in H-2 x R invariant by a vertical translation T, called saddle towers, which have total intrinsic curvature 4 pi(1 - k), genus zero and 2k vertical Scherk-type ends in the quotient by T. Each of those examples is obtained from the conjugate graph of a Jenkins-Serrin graph over a convex polygonal domain with 2k edges of the same (finite) length. As limits of saddle towers, we obtain properly embedded minimal surfaces, called minimal k-noids, which are symmetric with respect to a horizontal slice (in fact they are vertical bi-graphs) and have total intrinsic curvature 4 pi(1 - k), genus zero and k vertical planar ends.