Application of Scans and Fractional Power Integrands

In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Rivière and the second author to adequately describe certain bubbling phenomena. There, the behavior of certain W 1,3 weakly convergent sequences of smooth maps from 4 dimensional domains into S led to the consideration of a necessarily infinite mass generalization of a rectifiable current. The definition of a scan is motivated by the fact that a rectifiable current can be expressed in terms of its lower dimensional slices by oriented affine subspaces. By integral geometry, the slicing function for the rectifiable current is a mass integrable function of the subspaces. With a scan one considers more general such functions that are not necessarily mass integrable.