Seminar Analysis: Annalisa Massaccesi (University of Zurich)

Frobenius property for integral currents and decomposition of normal currents

In this joint work with Giovanni Alberti, we prove a Frobenius property for inte-gral currents: namely, if R=[∑,ξ,θ] k-dimensional integral current with a simple tangent vector field ξ∈C¹(R^d;Λ_k(R^d)), then ξ is involtive at almost every point in ∑. This result is related to the following decomposition problem formulated by F.Morgan: given a k-dimensional normal current T, do there exist a measure space L and a family of rectifiable currents {R_λ}_λ∈L such that T = ∫_L R_λ dλ and the mass decomposes consistently as M(T) = ∫_L M(R_λ) dλ? The aforementioned Frobenius property allows us to provide a counterexample to the existence of such a decomposition with a family of integral currents.

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