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