Integration of differential forms on manifolds with locally finite variations

A. V. Potepun

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 168-206

Résumé

It is well known that one can integrate any compactly supported continuous differential $n$-form over $n$-dimensional $C^1$-manifolds in $\mathbb R^m $ ($m\ge n$). For $n=1$ the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported $C^\infty$-differential forms). The theme of the article is integration of measurable differential $n$-forms over $n$-dimensional $C^0$-manifolds in $\mathbb R^m$ with locally finite $n$-dimensional variations (a generalization of locally rectifiable curves to dimension $n>1$). The main result states that any such manifold generates an $n$-dimensional current in $\mathbb R^m$ (i.e., any compactly supported $C^\infty$ $n$-form may be integrated over a manifold with the properties mentioned above).