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 \(\mathbb {R}^2\) ${\mathbb{R}}^2$ . As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ .