Motivated by the work of K. Uhlenbeck, I discuss minimal immersions of closed surfaces of genus larger than 1 on hyperbolic 3-manifold. In this respect we establish multiple existence for the Gauss –Codazzi equation and describe the asymptotic behaviour of the solutions in terms of the marked conformal structure on the surface and the (prescribed) second fundamental form of the minimal immersion. Joint work with Z. Huang and M. Lucia.