Density results relative to the Dirichlet energy of mappings into a manifold

Let be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps u k : B n → with an equibounded Dirichlet integral give rise to elements of the space cart 2,1 ( B n × ). Assume that is 1‐connected and that its 2‐homology group has no torsion. In any dimension n we prove that every element T in cart 2,1 ( B n × ) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u k : B n → with Dirichlet energies converging to the energy of T . © 2006 Wiley Periodicals, Inc.