New Examples of Constant Mean Curvature Surfaces in $\mathbbS^2\times\mathbbR$ and $\mathbbH^2\times\mathbbR$

We construct nonzero constant mean curvature H surfaces in the product spaces S2 × R and H2 × 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 S2 × R for any H > 0 (some of which are compact) and in H2 × R for any H > 1/2 (which are shown to be properly embedded bigraphs). Finally, we give a different construction in H2 × R for H = 1/2, giving surfaces with the symmetries of a tessellation of H2 by regular polygons.