In a seminal paper, Uhlenbeck studied the set of first and second fundamental form that arise from minimal immersions of a given surface into a three dimensional hyperbolic manifold. However, as will be explained in this talk, such data are not suitable for describing the moduli space of minimal immersions. Following the ideas of Goncalves-Uhlenbeck, one must instead consider the “dual data” given by the cohomology class of (0,1) forms. To justify that approach, one is led to justify the existence and uniqueness of critical points of a functional whose lack of regularity and compactness do not allow to apply classical variational results. This is a joint work with Z. Huang and G.Tarantello.