  It is known that a complete immersed minimal surface with finite total
curvature in $\mathbb H^2\times\mathbb R$ is proper, has finite topology and
each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth
and Rosenberg, 2006; Hauswirth, Nelli, Sa Earp and Toubiana, 2015). In this
paper we prove that these three properties characterize complete immersed
minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$.
As corollaries of this theorem we obtain characterizations for minimal
Scherk-type graphs and horizontal catenoids in $\mathbb H^2\times\mathbb R$. We
also prove that if a properly immersed minimal surface in $\widetilde{\rm
PSL}_2(\mathbb{R},\tau)$ has finite topology and each one of its ends is
asymptotic to a geodesic polygon at infinity, then it must have finite total
curvature.
