For a closed embedded surface in the round 3-sphere, one can consider its Gauss map taking values in the Grassmannian of 2-planes in 4-space, and try to minimize its area among all surfaces of a given genus. We will present some motivations to study this functional, its relation to other variational problems, and a dichotomy between spheres, for which the infimum is achieved, and surfaces of genus g>0, for which there are no minimizers and minimizing sequences must degenerate. After describing some interesting ways in which this degeneration can occur, giving rise to a round sphere and g-1 small handles where the negative curvature concentrates, we will present a compactness result showing roughly that all minimizing sequences exhibit that behaviour. This is based on joint work with T. Riviere.