A structure property of “vertical” integral currents, with an application to the distributional determinant

We deal with integral currents in Cartesian products of Euclidean spaces that satisfy a “verticality” assumption. The main example is the boundary of the graph of some classes vector-valued and non-smooth Sobolev maps, provided that the boundary current has finite mass. In fact, the action of such currents is non-zero only on forms with a high number (depending on the Sobolev regularity) of differentials in the direction of the vertical space. We prove that such vertical currents live on a set that projects on the horizontal space into a nice set with integer dimension. The dimension of the concentration set is related to the level of verticality that is assumed. Therefore, for boundary of graphs of Sobolev maps, this dimension decreases as the Sobolev exponent increases. As an application, we then prove a concentration property concerning the singular part of the distributional determinant and minors.