On the harmonic maps from ${\bf R}\sp 2$ into $H\sp 2$

In this paper, we prove that normalized harmonic maps from ${{\mathbf{R}}^2}$ or ${{\mathbf{R}}^2}\backslash \{ 0\}$ into ${H^2}$ are just geodesies on ${H^2}$ and that the quasiconformal harmonic maps from ${{\mathbf{R}}^2}$ into ${H^2}$ are constant maps. We prove also that the only solution to $\Delta \alpha = \sinh \alpha \,$ on ${{\mathbf{R}}^2}\backslash \{ 0\}$ is the zero solution.