  Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian
3-manifold. We consider incompressible maps from $S$ to $M$ that are critical
points of an energy functional $F$ which is homogeneous of degree $1$. These
"minimizing" maps are solutions of a non-linear elliptic equation, and
reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing
maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in
$M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a
given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps from
$(S,h)$ are described by a simple holomorphic data on $S$: a complex
self-adjoint Codazzi tensor of determinant $1$. The space of admissible data is
smooth and naturally equipped with a complex structure, for which the monodromy
map taking a data to the holonomy representation of the image is holomorphic.
Minimizing maps are in this way reminiscent of shear-bend coordinates, with the
complexification of $F$ analoguous to the complex length.
