On the harmonic maps from $\textbf {R}^ 2$ into $H^ 2$

Abstract:

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.