Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold

Let (S, h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We con-sider 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 mini-mizing 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 monodromy 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.