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Some new results about the geometry of sets of finite perimeter

Part of: Classical measure theory Manifolds

Published online by Cambridge University Press:  23 December 2015

Silvano Delladio
Silvano Delladio*
Affiliation:
Department of Mathematics, University of Trento, via Sommarive 14, Povo, 38123 Trento, Italy (silvano.delladio@unitn.it)
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Abstract

We establish that the intrinsic distance dE associated with an indecomposable plane set E of finite perimeter is infinitesimally Euclidean; namely, in E. By this result, we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential. We also provide some applications to complex analysis. Moreover, we present a collection of results that would seem to suggest the possibility of developing a De Rham cohomology theory for integral currents.

Keywords

sets of finite perimetertopology and geometry of setsinfinitesimal Euclideanity

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory 49Q15: Geometric measure and integration theory, integral and normal currents
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 1 , February 2016 , pp. 79 - 105
Copyright
Copyright © Royal Society of Edinburgh 2016 

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