Foliated plateau problem,part II: Harmonic maps of foliations

Our basic results concerning harmonic maps are parallel to those in Part I [Gro11] about minimal subvarieties. First we produce compact harmonic foliations by solving in some cases the asymptotic Dirichlet problem. Then we construct transversal measures by adopting the parabolic equation method of Eells and Sampson. Finally we indicate some applications to the rigidity and the pinching problems.