Simply Connected Minimal Surfaces with Finite Total Curvature in ℍ2 × ℝ

Laurent Hauswirth and Harold Rosenberg developed in [5] the theory of minimal surfaces with finite total curvature in $\mathbb H^2\times \mathbb R$. They showed that the total curvature of one such surface must be a nonnegative integer multiple of −2π. The first examples appearing in this context are vertical geodesic planes and Scherk minimal graphs over ideal polygonal domains. Other nonsimply connected examples have been constructed recently in [9, 12, 17]. In this paper, we show that the only complete minimal surfaces in $\mathbb H^2\times \mathbb R$ of total curvature −2π are Scherk minimal graphs over ideal quadrilaterals. We also construct properly embedded simply connected minimal surfaces with total curvature −4kπ, for any integer k≥1, which are not Scherk minimal graphs over ideal polygonal domains.