Calculus of Variations and Geometric Measure Theory
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G. Alberti - A. Massaccesi

On some geometric properties of currents and Frobenius theorem

created by alberti on 28 May 2017
modified on 04 Oct 2017


Accepted Paper

Inserted: 28 may 2017
Last Updated: 4 oct 2017

Year: 2017

ArXiv: 1705.09938 PDF


In this note we announce some results, due to appear in 2, 3, on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in 1.

Keywords: integral currents, Sobolev surfaces, non-involutive distributions, Frobenius theorem, decomposition of normal currents, foliations


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