On sequence of maps with equibounded energies
Let \(X^n, Y^m\) be oriented Riemannian manifolds, with \(Y^m\) compact and such that its second homology group has no torsion. The main result of the paper is a closure theorem for a class of Cartesian currents in \(X^n\times Y^m\), extending to any dimension results previously known only for \(n\leq 3\). A suitable class of Cartesian currents is introduced, by looking at the properties of generic 2-dimensional slices of them, and one of the main results of the paper is that in the decomposition \[ T=G_u+\sum_{i=1}^s L_i\times\gamma_i, \] where \(G_u\) is the current associated to the non-vertical part of \(u\) and \(\gamma_i\) are generators of the \(2\)-homology group of \(Y\) with integer coefficients, \(L_i\) are rectifiable \((n-2)\)-currents. The proof is achieved through a dimension reduction argument and uses characterizations of rectifiability through lower dimensional slices proved by \textit{B. White} [Ann. Math. (2) 150, No.~1, 165-184 (1999; Zbl 0965.49024)] in the integer case, and by \textit{L. Ambrosio} and \textit{B. Kirchheim} [Acta Math. 185, No.~1, 1-80 (2000; Zbl 0984.49025)] in the general rectifiable case. As a byproduct, for Cartesian currents minimizing a Dirichlet energy it is shown that the energy can concentrate only on \((n-2)\)-dimensional rectifiable subsets.
scientific article; zbMATH DE number 1655728