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.
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Communicated by A. Neves.
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Research partially supported by the MCyT-FEDER research project MTM2014-52368-P, MTM2017-89677-P, and by the GENIL research project no. PYR-2014-21 of CEI BioTic GRANADA.
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Hauswirth, L., Menezes, A. & Rodríguez, M. On the characterization of minimal surfaces with finite total curvature in \({\mathbb {H}}^2\times {\mathbb {R}}\) and \(\widetilde{\mathrm{PSL}}_2 ({\mathbb {R}})\). Calc. Var. 58, 80 (2019). https://doi.org/10.1007/s00526-019-1505-4
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DOI: https://doi.org/10.1007/s00526-019-1505-4