Minimal ends in H2xR with finite total curvature and a Schoen type   theorem
  In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal complete end E with finite total curvature is properly immersed and that the Gaussian curvature of E is locally bounded in terms of the geodesic distance to its boundary. 
Laurent Hauswirth, Barbara Nelli, Ricardo Sa Earp, Eric Toubiana
math.DG math.CV