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Minimising currents and the stable norm in codimension one. (English. Abridged French version) Zbl 1008.53058
Let $T$ 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 $$M_U(T)= \sup\bigl\{T(w): w\in\Omega_0^{n-1}(U),\ \|w\|_\infty\le 1\bigr\}.$$ $T$ is called locally minimizing if every point $x\in M$ has a neighborhood $U$ such that $M_U(T)\le M_U(T+S)$ 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 $M$.

53C65Integral geometry
58A25Currents (global analysis)
49Q15Geometric measure and integration theory, integral and normal currents (optimization)
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