  Given an exterior domain $\Omega$ of $C^{2,\alpha}$ class of
$\mathbb{R}^{n},$ $n\geq3,$ it is proved the existence of a foliation of the
closure of an open set of
$\mathbb{R}^{n+1}\backslash\left(\partial\Omega\times\mathbb{R} \right) $ which
leaves are graphs of the solutions of the exterior Dirichlet problem on
$\Omega$ for the minimal surface equation containing the trivial solution$.$
The leaves have horizontal ends and are parametrized by the maximum angle
$\alpha\in\left[ 0,\pi/2\right] $ that the Gauss map in $\mathbb{R}^{n+1}$ of
the leaves make with the vertical direction at the boundary of the domain. The
solutions have a limit $c_{\alpha}$ at infinity, the map $\alpha\mapsto
c_{\alpha}$ is increasing and bounded. Moreover, if
$\mathbb{R}^{n}\backslash\Omega$ satisfies the interior sphere condition of
maximal radius $\rho$ and if $\partial\Omega$ is contained in a ball of minimal
radius $\varrho$ then \[ \left[ 0,\sigma_{n}\rho\right] \subset\left[
0,c_{\pi/2}\right] \subset\left[ 0,\sigma_{n}\varrho\right] , \] where \[
\sigma_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}. \] One
of the above inclusions is an equality if and only if $\rho=\varrho,$ $\Omega$
is the exterior of a ball of radius $\rho$ and the solutions are radial.
  These foliations were studied by E. Kuwert (1993) and our result answers a
natural question about the existence of such foliations which was not touched
on his work.
