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Minimising currents and the stable norm in codimension one. (English) 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)=supT (w) : w Ω 0 n-1 (U) , w 1·

T is called locally minimizing if every point xM has a neighborhood U such that M U (T)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 M ¯ of M.

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