The Dirichlet integral for mappings between manifolds: Cartesian currents and homology (Q1195982)

The paper deals with the Dirichlet integral for maps between two Riemannian manifolds \(X\) and \(Y\) and develops a kind of homological theory for the Dirichlet integral. Identifying maps with graphs, i.e. currents, first, limits of smooth graphs with equiconnected Dirichlet integrals are studied. This leads to define a suitable subclass of Cartesian currents. The structure of such currents then fully discussed in the case \(\dim X=2,3\) and \(H_ 2(Y,\mathbb{Z})\) is non-empty and torsionless, in particular it is proved that concentrations of energy defined on the cycles in \(H_ 2(Y,\mathbb{Z})\) of type \(S^ 2\) and may occur only on one-dimensional rectifiable sets (if \(\dim X=3)\). The Dirichlet integral is then extended to such currents and explicitely computed. Finally, variational problems for maps with prescribed two-homology maps, or with prescribed singularities are discussed.