Motion of level sets by mean curvature IV

ABSTRACT. We continue our investigation of the "level-set " technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact a unit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of "geometric structure. " The proof utilizes compensated compactness methods to pass to limits in various geometric expressions. I