Periodic constant mean curvature surfaces in H² x R

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In this paper we study minimal and constant mean curvature (cmc) periodic surfaces in H² x R. More precisely, we consider quotients of H² x R by discrete groups of isometries generated by horizontal hyperbolic translations f and/or a vertical translation T. In the quotient by the Z² subgroup of the isometry group generated by any f and T, we prove an Alexandrov-type theorem for cmc surfaces, i.e., an analysis of compact embedded cmc surfaces in such a quotient. The remainder of the paper is devoted to construct examples of periodic minimal surfaces in H² x R.