Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
Functional Analysis
2013-04-01 v2 K-Theory and Homology
Metric Geometry
Abstract
We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.
Keywords
Cite
@article{arxiv.1105.1995,
title = {Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket},
author = {Fabio Cipriani and Daniele Guido and Tommaso Isola and Jean-Luc Sauvageot},
journal= {arXiv preprint arXiv:1105.1995},
year = {2013}
}
Comments
Some proofs have been clarified, reference to previous literature is now more accurate, 33 pages, 6 figures