  We develop a conformal duality for spacelike graphs in Riemannian and
Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian
surface whose fibers are the integral curves of a Killing vector field, which
is timelike in the Lorentzian case. The duality swaps mean curvature and bundle
curvature and sends the length of the Killing vector field to its reciprocal
while keeping invariant the base surface. We obtain two consequences of this
result. On the one hand, we find entire graphs in Lorentz-Minkowski space
$\mathbb{L}^3$ with prescribed mean curvature a bounded function $H\in
C^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain
conditions for existence and non existence of entire graphs which are related
to a notion of critical mean curvature.
