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Laurent Hauswirth and Harold Rosenberg developed the theory of minimalsurfaces with finite total curvature in $\H^2\times\R$. They showed that thetotal curvature of one such a surface must be a non-negative integer multipleof $-2\pi$. The first examples appearing in this context are vertical geodesicplanes and Scherk minimal graphs over ideal polygonal domains. Other nonsimply-connected examples have been constructed recently. In the present paper,we show that the only complete minimal surfaces in $\H^2\times\R$ of totalcurvature $-2\pi$ are Scherk minimal graphs over ideal quadrilaterals. We alsoconstruct properly embedded simply-connected minimal surfaces with totalcurvature $-4k\pi$, for any integer $k\geq 1$, which are not Scherk minimalgraphs over ideal polygonal domains.

Simply-connected minimal surfaces with finite total curvature in $\H^2\times\R$