### ... Minimizing Currents.

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For vector valued maps, convergence in ${W}^{1,1}$ and of all minors of the Jacobian matrix in ${L}^{1}$ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension $n\ge 3$ can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain.

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi}_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^{m}$-measurable subset of ℝⁿ with ${}^{m}\left(A\right)<\infty $. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi}_{V}\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for ${}^{m}$-a.e. P ∈...

This paper gives a new proof of the fact that a $k$-dimensional normal current $T$ in ${\mathbb{R}}^{m}$ is integer multiplicity rectifiable if and only if for every projection $P$ onto a $k$-dimensional subspace, almost every slice of $T$ by $P$ is $0$-dimensional integer multiplicity rectifiable, in other words, a sum of Dirac masses with integer weights. This is a special case of the Rectifiable Slices Theorem, which was first proved a few years ago by B. White.