On doubly periodic minimal surfaces in H²× R with finite total curvature in the quotient space

In this paper, we develop the theory of properly immersed minimal surfaces in the quotient space (H²× R) / G, where G is a subgroup of isometries generated by a vertical translation and a horizontal isometry (without fixed points) in H². The horizontal isometry can be either a parabolic translation along horocycles in H² or a hyperbolic translation along a geodesic in H². We prove that if a properly immersed minimal surface in (H²× R) / G has finite total curvature, then its total curvature is a multiple of 2 π and, moreover, we understand the geometry of the ends. The results hold true more generally for properly immersed minimal surfaces in M× S¹, where M is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces (H²× R) / G.