  The main aim of this survey paper is to gather together some results
concerning the Calabi type duality discovered by Hojoo Lee between certain
families of (spacelike) graphs with constant mean curvature in Riemannian and
Lorentzian homogeneous 3-manifolds with isometry group of dimension 4. The
duality is conformal and swaps mean curvature and bundle curvature, and we will
revisit it by giving a more general statement in terms of conformal immersions.
This will show that some features in the theory of surfaces with mean curvature
$\frac{1}{2}$ in $\mathbb{H}^2\times\mathbb{R}$ or minimal surfaces in the
Heisenberg space have nice geometric interpretations in terms of their dual
Lorentzian counterparts. We will briefly discuss some applications such as
gradient estimates for entire minimal graphs in Heisenberg space or the
existence of complete spacelike surfaces, and we will also give an uniform
treatment to the behavior of the duality with respect to ambient isometries.
Finally, some open questions are posed in the last section.
