Minimising currents and the stable norm in codimension one.

*(English)*Zbl 1008.53058Let $T$ be a closed current of dimension $(n-1)$ on the $n$-dimensional Riemannian manifold $M$. Suppose that $T$ is of locally finite mass. Recall that for an open $U<M$ the mass of $T$ in $U$ is defined as

$${M}_{U}\left(T\right)=sup\left\{T\left(w\right):w\in {{\Omega}}_{0}^{n-1}\left(U\right),\phantom{\rule{4pt}{0ex}}{\parallel w\parallel}_{\infty}\le 1\right\}\xb7$$

$T$ is called locally minimizing if every point $x\in M$ has a neighborhood $U$ such that ${M}_{U}\left(T\right)\le {M}_{U}(T+S)$ for any closed current with locally finite mass $S$ supported in $U$. The authors prove that every locally minimizing current is given in fact by a lamination by singular minimal hypersurfaces on an appropriate covering $\overline{M}$ of $M$.

Reviewer: Sergey M.Ivashkovich (Villeneuve d’Ascq)