quasiconformal mappings and a Bernstein type theorem over exterior domains in ${\mathbb{R}}^2$

We establish the Hölder estimate and the asymptotic behavior at infinity for $K$-quasiconformal mappings over exterior domains in ${\mathbb{R}}^2$. As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in ${\mathbb{R}}^2$.