New Examples of Constant Mean Curvature Surfaces in S2 x R and H2 x R

We construct nonzero constant mean curvature H surfaces in the product spaces S-2 x R and H-2 x R by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height, and are invariant under a discrete group of horizontal translations. A one-parameter family of unduloid-type surfaces is produced in S-2 x R for any H > 0 (some of which are compact) and in H-2 x R for any H > 1/2 (which are shown to be properly embedded bigraphs). Finally, we give a different construction in H-2 x R for H = 1/2, giving surfaces with the symmetries of a tessellation of H-2 by regular polygons.