  We prove that any complete, uniformly elliptic Weingarten surface in
Euclidean $3$-space whose Gauss map image omits an open hemisphere is a
cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman
and Schoen for constant mean curvature surfaces. In particular, this proves
that planes are the only complete, uniformly elliptic Weingarten multigraphs.
We also show that this result holds for a large class of non-uniformly elliptic
Weingarten equations. In particular, this solves in the affirmative the
Bernstein problem for entire graphs for that class of elliptic equations. To
obtain these results, we prove that planes are the only complete multigraphs
with quasiconformal Gauss map and bounded second fundamental form.
