On the characterization of minimal surfaces with finite total curvature in $${\mathbb {H}}^2\times {\mathbb {R}}$$ H 2 × R and $$\widetilde{\mathrm{PSL}}_2 ({\mathbb {R}})$$ PSL ~ 2 ( R )

Abstract

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 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 $${\mathbb {H}}^2\times {\mathbb {R}}$$ . 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 $$\widetilde{\mathrm{PSL}}_2({\mathbb {R}},\tau )$$ has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.