A Construction of Constant Mean Curvature Surfaces in H-2 x R and the Krust Property

We show the existence of a 2-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature 0 <= H <= 1/2 in H-2 x R. They are symmetric with respect to a horizontal slice and k vertical planes disposed symmetrically and extend the so-called minimal saddle towers and k-noids. We show that the orientation plays a fundamental role when H > 0 by analyzing their conjugate minimal surfaces in (SL2) over tilde (N) or Nil(3). We also discover new complete examples that we call (H, k)-nodoids, whose k ends are asymptotic to vertical cylinders over curves of geodesic curvature 2H from the convex side, often giving rise to non-embedded examples if H > 0. In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any H > 0, that is, there are minimal graphs over convex domains in (SL2) over tilde (R), Nil(3) or the Berger spheres, whose conjugate surfaces with constant mean curvature H in H-2 x N are not graphs.