On the characterization of minimal surfaces with finite total curvature in H ² × R and PSL ~ ₂ (R)

It is known that a complete immersed minimal surface with finite total curvature in H ² × R is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg in Mat Contemp 31:65–80, 2006; Hauswirth et al. in Adv Math 274:199–240, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in H ² × R. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in H ² × R. We also prove that if a properly immersed minimal surface in PSL ~ ₂ (R, τ) has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.