Minimising currents and the stable norm in codimension one (Q2778035)

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 NEWLINE\[NEWLINEM_U(T)= \sup\bigl\{T(w): w\in\Omega_0^{n-1}(U),\;\|w\|_\infty\leq 1\bigr\}.NEWLINE\]NEWLINE \(T\) is called locally minimizing if every point \(x\in M\) has a neighborhood \(U\) such that \(M_U(T)\leq M_U(T+S)\) 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 \(M\).